General theory of doubly periodic arrays over an arbitrary finite field and its applications

IEEE TRANSACI-IONS ON INFORMATION THEORY, VOL. 1~24, NO. 6, NOVEMBER 1978 719 General Theory of Doubly Periodic Arrays over an Arbitrary Finite Field and Its Applications SHOJIRO SAKATA Abstract-A general theory of doubly periodic (DP) arrays over an arbitrary finite field GF(q) is presented. Fit the basic properties of DP arrays are examhed. Next modules of linear recurhg (LR) arrays are defined and their algebr$c properties discus& in connection with ideals in an extension ring R of t& ring R of bivariate polyuomials with coefficien@ iu GF(q). A finite R-module of DP arrays is shown to coinci@ with the R-module of LR arrays defined by a zero-dimensional ideal in R. Equivalence relations between DP arrays are explored, i.e., rearrangements of arrays by means of unimodular transformations. Decimation and interleaving of arrays are defined in a two-dimensional sense. The general theory is followed by application to irreducible LR arrays. Among irreducible arrays, M-arrays are a two-dimensional analog of M-sequences and may be constructed from M-sequences by means of unimodular transformations. The results of this paper are also important in studying properties of Abeliau codes. I. INTRODUCTION OUBLY periodic (DP) arrays over an arbitrary finite field GF(q) are a two-dimensional generalization of periodic sequencesover GF(q). Linear recurring sequenceswhich are generated by a linear shift register with feedback are eventually periodic sequences.A linear recurring sequence is determined, given a set of initial values, by a linear recurring relation or by a homogeneous difference equation which can be implemented with a linear feedback shift register. Our investigation mostly concerns a general property of purely periodic arrays, i.e., the general solutions for systems of homogeneous linear partial difference equations. It has not been mentioned by other authors on linear recurring (LR) arrays or planes [l]-[4] that the “period” of a DP array is not completely characterized by the partial periods perx (u) and perv (u). We should also compare DP arrays with elliptic functions, which are doubly periodic on the complex plane [8]. This approach is especially important in studying twodimensional cyclic codes in connection with LR arrays. It is similarly possible to investigate multidimensional periodic arrays and multidimensional cyclic codes. Multidimensional cyclic codes are also called Abelian codes, D Manuscript received August 1, 1977; revised April 18, 1978. Part of the results described in this paper were presented at conferences of the Technical Group on Automata and Languages, Institute of Electronics and Communications Engineers of Japan, December 1976 and March 1977, VI, 161. The author is with the Department of Information Science, Faculty of Engineering, Sagami Institute of Technology, Fujisawa 251, Japan. which are a natural generalization of cyclic codes. An Abelian code is defined to be an ideal in an Abelian group ring. Although several authors [9]-[13] have observed that many properties of cyclic codes carry over to Abelian codes, some intrinsic properties remain for further investigations. A general theory of two-dimensional cyclic codes has been proposed by Ikai et al. [14]-[16], where a “quasi-division” algorithm in the bivariate polynomial ring is introduced for the purpose of determining the positions of check symbols as well as generator polynomials. This technique has also been applied to develop a theory of two-dimensional LR arrays [3]. It will be shown in the present paper, however, that a more general approach is possible without invoking such a special device. Our method is based only on the fundamental concepts and theorems of ideals and modules and is therefore of wide applicability. It will also be important in devising more efficient encoding and decoding methods for Abelian group codes. Irreducible LR arrays are DP arrays of which the characteristic ideal Q is prime. They have the simplest algebraic structure and can be treated by a special device: the algebraic set I’(!@), i.e., the zeros belonging to the ideal. M-arrays are the most important irreducible LR arrays. They are two-dimensional analogs of M-sequences or maximum length sequences.yp-arrays have been introduced in [2] and have what is called the maximum area property. In the present paper a general treatment of DP and LR arrays will be presented. The basic mathematical properties are studied using the theory of modules over the ring R” which is isomorphic to an extension ring of the bivariate polynomial ring R. Correspondence relations between DP arrays are clarified by means of the automorphisms of the ring R” and R”-isomorphisms between Rmodules. The outline of the paper is as follows. Section II is devoted to preliminaries on DP and LR arrays. Section III describes the fundamental theorem which clarifies the relation between DP and LR arrays. The algebraic properties of LR arrays are explored in connection with their characteristic ideals. The independent point set, i.e., the positions of the information symbols in the terminology of algebraic coding theory, is determined in general. Section IV describes various correspondencesbetween DP arrays OO18-9448/78/1100-0719$00.7501978 IEEE 720 IEEE TRANSACTIONS ON INFORMATION THEORY. VOL. IT-B, determined by rearrangement, decimation, and interleaving, which are associated with automorphisms and endomorphisms of R”.In Section V the general treatment of DP and LR arrays will be applied to irreducible LR arrays. In a two-dimensionally proper sensewe will define M-arrays, which are shown to coincide with y&arrays and to be constructed from M-sequences by rearrangements. In this paper we make free use of concepts and theorems from commutative ring theory (e.g., [17], [18]) and matrix theory. II. NO. 6, NOVEMBER 1978 u . . . . . . . . . . 1011011... 1101101... 0110110... 1011011... 1101101... 0110110... 1011011... - ......... \ ........ ......... ......... \ ........ ......... ......... ......... ......... j T .......... .......... Fig. 1. Example of DP array over GF (2). u PRELIMINARIES The two-dimensional integral lattice is the totality of lattice points which have coordinates (ij) E Z2( b Z X Z), where Z is the ring of integers. A two-dimensional array U=(Q) is an arrangement of elements uiJ of F A GF(q) on the lattice points, where q is a power of a primep. The value uij of an array u on the point i= (iJ) E Z2 is said to be the (iJ)-component of u and is denoted alternatively by (u)~,. or by u(i). A two-dimensional array may be identified with a double sequence {ui,.} with values in F or an infinite matrix U over t;: U= ]]+]I. We investigate periodic arrays in the following sense. They are a discrete analog of periodic functions with two indeterminates [8]. Definition I: If an array u satisfies the relation u(z)= u(i + r), for some 1E Z2 and all i E Z2, the integral vector 1 is called a period (P.) vector of u. An array with a P. vector other than the zero vector 0 is called a periodic array. The set P of all P. vectors of u forms an additive group contained in Z2. If a basis of this additive group exists, its members are called fundamental period (FP) vectors of u. An array with two FP vectors is called a doubly periodic array. (SeeFig. 1.) Let I,, 1, be the FP vectors of a DP array. For a given vector 1E Z 2, the parallelogram with vertices 1, I + I,, I+ I,, I+ I, + 1, (or the set 1,X 1, of lattice points expressed as I+ w,l, + 0212, for rational numbers wi, w2 such that 0 < w,, w2< 1) is called the fundamental period parallelogram. The two-dimensional plane (or integral alttice) is covered with a network of congruent parallelograms, called period parallelograms, obtained by translating the FP parallelogram through ml, + nl,, m, n E Z. For a basis l,,l, of the additive group P, another basis m,,m2 is obtained from any unimodular matrix S, i.e., an integral matrix with the determinant det (S) = 1 or - 1, as follows: (1) . . . . . . . . . :0100010... .0011110... 0001010... :1111001... .0101000... .1100111... .0100010... . . . . . . . . . . . . . . . . . . . . . . &+-1x& qx “-2 0 0 01 0 001 110 10 00 11 10 1 11 1 0 Fig. 2. Example of DP array over GF (2) and its FP parallelograms. usual way. Then the set W of all DP arrays forms an F-module, i.e., a linear space over the field F. The identity element of the module is the zero array (0). For two submodules U, I’ of W, U+ V is the submodule of W which is composed of the arrays of the form u+ v for uEUandvEV. For some vector k =(k,I) E Z2 and an array u, the array u’ such that u’(i) = u(i + k), i E Z2, is said to be the translate of u along k and is denoted by ku or k,,u. There are a finite number of different translates of a DP array u. Let the cycle of u denote the set of all different translates of 24. Definition 2: The cardinality of the cycle of a DP array u is called the period of u and is denoted by per (u). per (u) is identical with the number of lattice points contained in a FP parallelogram I, X l2 of u and is also equal to the absolute value of 4 I det (ii 2 . II) A DP array has a P. vector with an arbitrary direction 1E Z2. The length of a FP vector with the direction of with a FP parallelogram 1,x I,, where I, =(4,2), 1,=(2,4). ix g (1,O) is said to be the partial x-period and is denoted For example, v also has a FP parallelogram m, X m,, by per, (u). The partial y-period per,, (u) is defined in the where m, = (4,2), m2= (6,6). They are shown in Fig. 2. same way. If per, (u) or perY (u) = 1, the array is the The sum u + v of two arrays u, v and scalar multiplica- perpendicular repetition of a (simple) periodic sequence. tion CMof an array u by a constant are defined in the This is the simplest case of double periodicity. Example I: Consider a DP array v over GF(2) = (0, 1} 721 SAKATA: DOUBLY PERIODIC ARRAYS A DP array satisfies the trivial linear recurring relations .= u. ‘) uij+, = uiJ, (i,j) E Z2, for integers r, s such that ~e:J(u)$( per (u)l s.’ In order to deal with LR relations in general, weYintroduce polynomial-like operators f(x,y) =~:(I,J)El-Cffi,,JXIYJ~where P(j) is a finite subset of Z2 andf,,, are elementsof F. Byf(x,y)u we mean the array v such that vij = Z (I,J)El&,JUi+I,j+J~ (i,j) E Z 2. In particular, x%=,,,u and y/u= a Iu are translates of u. These operators form a ring, which is isomorphic to the ring R” k {x$‘f(x,y)li,jEZ, f(x,y)ER}, where R is the ring of bivariate polynomials with coefficients in F. From now on w,e do not distinguish between the ring of operators and R. Definition 3: For an ideal ‘% in the ring R”, let G(8) denote the set of all arrays u such that, for any element f(X>Y) of % f(x&Y)u = (0). G(X) is an R-module, i.e., a module which is closed under any operations of I?. Any ideal % in R has a finite basis !X=cf(i); - ’ ,f(M)). (See Appendix A.) Clearly G(g) coincides with the set of linear recurring arrays u which satisfy the LR relations f@)(x,y)u = (0), 1 <k <M, or equivalently, for f@)(.x,y) = ~:(,,~)e-(~)f~l;x+‘> 1 <k GM, pith the finite algebraic set V(%),3if and only if U is an R-module of DP arrays. We begin with treating the i-module G(z) of LR arrays. Lemma 4: Provided that $I is a zero-dimensional ideal in R”, G(8) is a finite subset of the set W of all DP arrays. Proof We may choose polynomials f(l), * * - ,fcM) forming a basis of ‘9I. Eliminating y from the system of algebraic equations f”‘(x,y)=o * * * ,f’“‘(x,y)=O we obtain the following iystem [17, pp. 1151: D(‘)(x)= 0; * * , D@‘)(x) = 0. Polynomials D (‘j(x) . . . , D@)(x) are elements of VI. Since !JI is zero-dimensional, for some k D(x)= D@)(x) does not vanish identically. D(x) can be assumedto have a nonzero constant term. Then there is a positive integer r such that 0(x)1x’1. Therefore %3x’ - 1. In the same way 8 3ys - 1 for a certain postive integer s. Thus any array u in G(%) is doubly periodic with partial periods such that per, (u)lr, pery (u)ls. Q.E.D. For a one-dimensional ideal % in R”, there exist arrays in G(z) which have a finite P. vector only along a single direction. From now on we exclude such cases, dealing E fj/YUi+I,j+J=07 (i,j> E Z2. (2) only with zero-dimensional ideals. A finite R-module (I,-‘) ET(k) G(8) is a finite-dimensional F-module. From Definition 3 we have immediately the following Definition 4: Let dim G(%) denote the dimension of the lemma. F-module G(E). Lemma I: If 9I and ‘93are ideals in R” such that XI !8, The linear recurrences (2) can be regarded as a system then G(%)c G(8). of linear equations with respect to the rs components uiJ, Provided that there exist integers r,s such that 9I 1 (xr O<i<r-l,O<j<s-1, - 1,~” - 1),2 the partial periods of $I [ 141are defined as r-l s-l follows: x c fjyJ)UiJ = 0, per, (%)=min {r>Olx’- i-0 1E%} *pery (%!I)=min {s>Oly”- I E%}. j=O O<Z<r-1, O<J<s-1, l<k<M (3) where Lemma 2: For an arbitrary array u in G(X), per, if (i-Z j-J)ET(k) fi(jk,IJ) = ~!~‘I,j-J~ othenv;se (u)lper, @I), perv (u)lper, @I), provided that % has finite L partial periods. and the subscripts iJ are to be interpreted respectively It is easy to see the validity of the following relation. r and mod s. The polynomials f(k”,J)(~,y) k mod Lemma 3: For the sum ??I+% of ideals 9I, !J3 in R”, y~y;$$JJ)x$j can be regarded as elements of the G(8) n G(B) = G(Ca + 23). residue class ring g/(x’l,y”- 1). Clearly, dim G(9I) is equal to the dimension of the solution space of the linear III. LINEAR RECURRING ARRAYS AND DOUBLY system (3). Since the rank of (3) is equal to the dimension PERIODIC ARRAYS dim, (g/(x’1,~” - 1)) of the residue class ring s/(x’1,~” - 1) as an F-module, we have from Lemma Bl (ApIn this section we give the complete characterization of pendix B), DP arrays as LR arrays. The following theorem ensures (8/(x’-l,y”-1)) that we have only to examine polynomial ideals in order dim G(%)=rs-dim, to investigate the properties of DP arrays. =dim, (i/(x’- Theorem 1 (Fundamental Theorem on Doubly Periodic and Linear Recurring Arrays): Let U be a finite nonempty =dim, (RI/%). set of arrays. U coincides with the set G(%j of LR arrays for a certain zero-dimensional ideal 9l in R, i.e., an ideal ‘alb means that there exists an integer c such that UC= b. 2Equivalently, % is a zero-dimensional ideal. (See Lemma 4.) l,y”- 1)) -dim, (91/(xr- l,y”- 1)) (4) 3The algebraic set V(‘%) is defined to be the totality of all zeros common to the polynomials contained in P. The smallest possible number of parameters which are needed in the parametric representation of an algebraic set P’(a) is called the dimension of V(a) or the dimension of P [l7, pp. 46-711. 122 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-24, NO. 6, NOVEMIJER 1978 It should be noted that (4) is valid irrespective of the w n lot) choice of integers r,s as long as 9 1 (xl’-- 1,~” - 1). Thus . . .. . . . .. i . . . . . . . . . . . the F-module G(%) is (F-) isomorphic to the F-module --l 1111001... :1110011... R”/2L 0111100... . \:r+ : : : : : : : The i-module G(Yl) is also an R”/%-module. More :0100010... . 1.1 . . . . . . . . 1000101... . . . . . . . . . generally we have the following lemma. :1010001... . . . . . . . . . .1111001... . . . . . . . . . Lemma 5: Provided that the ideal % is a subset of the . . . . . . . . . . . . . . . . . . . . ideal %, the difference module G(B)- G(X) is a a/‘%. . . . . . . . . . . . . . . . . . . . i module with its dimension as an F-module equal to that I=--of 9X/%% Fig. 3. Example of LR array over GF (2) and independent point set. If 8l z B, then dim G(E) < dim G(B), since R”/% s R/B. From this remark we have the converse to recursively by Lemma 1. Lemma 6: If G(‘%)c G(B), then %1%. (x2+y(x3+1)+x+1)w Proof: Suppose G(%) c G(B) and %a%. Then G(8) =(x$+(x2+ l)~)W = G(X) n G(%) = G(%+ !@. On the other hand, 9l+ ‘93z 9.L = (x2y2+y2(x + 1) +y(x3+ x2+ x))w from which it follows that G(g) 3 G(%+ !@. This is a =(y’+(x+1)y2+(x+1)y+x)w=(0), contradiction. Q.E.D. Let K= dim G(g). Then there is a unique array in G(%) which can be derived from (5). with an arbitrary set of values ujj in F for (i,j) in a certain For any finite set of DP arrays, there exists a corresubset r of 2’ such that ]I’] = i. Such a subset r of 2’ is sponding ideal. said to be an independent point set of the ideal % or of Theorem 3: If U is a finite nonempty subset of W, then G(IU) and is denoted by A(X). r may be obtained from a there exists a unique (zero-dimensional) ideal % in R such canonical form of the homogeneous linear system (3). that an ideal !?3is contained in % if and only if U is From the preceding considerations we have the following contained in G(‘%). The ideal % coincides with the set theorem. S(U) 22{f(x,y)E ilf(x,y)u =(O), u E U}. Theorem 2: A subset I? of Z2 is an independent point Definition 5: The ideal ‘%(U) is said to be the maximum set of a (zero-dimensional) ideal ?l if and only if ideal associated with U or the characteristic ideal of U. If i) (r( = dim G(g); U contains a single array U, a({ u}) is also said to be the ii) % does not contain a nonzero polynomial of the characteristic ideal a(u) of ua4 formf(x,y)=~(iJ3Erfi,jXLj. a(u) may be compared with the ideal generated by the minimum polynomial of a periodic sequence. As for the For (k,I)@A(!J), there is an element g@“)(x,y) of 9l partial periods of g(u), we have, in view of Lemma 1 and such that g(“,‘)(x,y) = x5’+ ZCi,jjEACPjg$‘)x5j.Thus for a the maximality of a(u) for U, the following lemma and its given set of values uij E F((ij) E A(%)), we can determine corollaries. the value uk,/ for any point (k,Z)@A(%) from Lemma 7: Let 8l = g(u); then perx (u) = per* (%), and uk,l= - ci,jlAca similarly for y . gi?“%* Corollary 1: per, (%)=max {per, (u)]u E G(g)}, and Note that an independent point set as defined by Ikai et similarly for y . al. [ 161is a special case of Theorem 2. Corollary 2: Provided that !B is a (zero-dimensional) prime ideal, for any nonzero array u in G(v), per, (u) = Example 2: Let q=2. The system of LR relations: perx (!B), and similarly fory. wi+dj+ wi+2,j+ wij=o Both perx (9) and pe’; (9) are relatively prime to the characteristic p of the field F. There is the following wi+2,j+l+wi+I,j+I+wi,j+l+wi+3j+wi+2,j+wi+l,j=o relation between a primary ideal Q and the prime ideal !J3 / wi,j+3+wi+l,j+2+wi,j+2+wi+I,j+I+wi,j+I+wi+I,j=o belonging to Q [ 17, pp. 26-301: for some integer I, !J31~9 (5) I!@‘. It follows that per, (?3)= perx ($31~m, per,, (a) = per,, (!$)p”, where m and n are inmgers such that 0 <p”,p” <I. corresponds to the ideal %=(x4 + x2 + 1, (x2 + x + l)(y + The sum G(‘%)+ G(!B) of R-modules G(%) and G(a) is x),y3+(x+l)y2+(x+l)y+x). For a root (Yof x2+x+1 an R-submodule of G(%n 8) and of G(%*!.@.The followin the extension field GF (22), the algebraic set is V(s)= ing result corresponds to that of Ikai et al. 13, th. 81. {(a,a),(a2,~2),(~,~2),(~2,~)}. A LR array w satisfying (5) Theorem 4: G(B) + G(B) = G(% n 8). is shown in Fig. 3, where perx (w)=per,, (w)=6, per (Ll), (19% G&l), tw)=36. WV= (64% (O,l>, (0,2), (l,O), (3, l)} is an independent point set and dim G('i?l)= 8. From 4A method of determining the characteristic ideal X(u) of a given DP the values in A(%), the remaining values are determined array u is described in Appendic C. 723 SAKATA: DOUBLY PERIODIC ARRAYS Proof: From Lemma B2 (Appendix B), we have dim, (8I + B/ %) = dim, @I/‘%n B), from which it follows, in view of Lemma 5, that dim G(B)-dim G(%+%J)=dim G(8 n 8) - dim G(g). Accordingly dim (G(g) + G(B)) = dim G(a) + dim G(B) -dim (G(g) n G(B)) = dim G(g) + dim G(B) -dim G(9.I+ 8) = dim G(X n 23), from which it follows that G(a) + G(B) = G(9l n 93). Q.E.D. Provided that two ideals %,!B are relatively prime, the k-module G($ n 9) = G(%. 8) is a direct sum of k-modules_G(8) and G(B), because G(8l)n G(B)= G(8+%)= G(R)= ((0)). A (zero-dimensional) ideal 9l can be represented as the intersection of greatest primary components (Appendix A): 8l= n jXi. Then the algebraic set is v(9.l)= u i V(~i) = u i I, where !j3, are distinct zero-dimensional prime ideals, each ~3,belonging to ai. Becaus; two distinct zero-dimensional prime ideals !I&,!$$ in R are relatively prime, the primary ideals 8.X1, %,, respectively, belonging to !J331,!J32, are also. Therefore, the above decomposition can be rewritten as 8= n ,9X,= II,& Corollary 3: G(%) can be expressedas the direct sum G(8) = G(i?I,)i . - . i G(iX,). Proof of Theorem I: Necessity is already trivial. Let Ul,“‘, uM be the elementsof U and ak be the characteristic ideal of each u,, 1 <k GM. Then G@l,) c U from Lemma 8 and G(%i) + . . . + G(%,) c U. But U c G(8,) + . . . + G(8,), since ukE G(8.1k),1 <k GM. Therefore, U = G@,) + . - . + G(%,)= G(s) from Theorem 4, where l%=%,n.-. n%,. Q.E.D. From Theorem 1 we also obtain the one-to-one correspondence between i-modules of DP arrays and two-dimensional cyclic codes, which are special casesof Abelian codes. IV. EQUIVALENT DOUBLY PERIODIC ARRAYS Given an integral unimodular matrix S, the array U’ such that u’(i) = u(iS), iE 2*, is said to be the array +s(u) rearranged from u by means of the unimodular transformation S. Lemma 9: For kS = m, $&,u) = k+s( u). Definition 6: If two arrays U,IJ are transformed into each other by rearrangement and/or translation, they are said to be equivalent to each other. If, an array u may be rearranged to give another array U’ by means of T, a P. vector 1 of u is transformed to a P. vector I’ of u’ by I’ = IT - ‘. Thus in view of (I), the following relation holds between the FP parallelogram Example 3: The ideal % in Example 2 can be decom- I, x f2 of u and the FP parallelogram m, X m2 of u’ e&(u): posed into greatest primary components as follows: %= %,*912,where 8,=(x2+x+l,y+x+1) is the prime ideal with I$lI,) = {( (~,(~~),(a’,a()}and ‘%,=(x4+x2+ 1, (x2+x (6) + 1)~+ x3 + x2 + x, y2 + x2) is a primary ideal belonging to the prime ideal Q2=(x2+ x + 1,~+x) with V(!X2)= a2,a2)}. The array w (Fig. 3) can be rep- where S and T are unimodular matrices. w32)= {(%4,( Lemma 10: A DP array has the same period as any resented uniquely by the sum of u (Fig. 1) in G(8,) and 0 array equivalent to it. The cycle containing u is trans(Fig. 2) in G(%,). 3, may be decomposed as 8, = ?3n iQ’, formed to the cycle containing an array equivalent to u. where ~=(x4+x2+1,y+x),~=(x2+x+l,y2+x2) are In view of (6), we have the following theorem. primary ideals belonging to q2. But G(Q)+ G(U) is not a direct sum. Theorem 5: An arbitrary DP array with a FP paralleloCorollary 4: Provided that X and Z! are relatively prime gram I, x I2 has an equivalent array of which a FP ideals, per, (U + v) = lcm (per, (u), perx (v)), for u E G(s), parallelogram I,’ x Ii is a rectangle with f; = (r, 0), 1; = (O,s), 1)E G(B), and similarly fo_ry. where r and s are elementary divisors of the integral We now treat a finite R-module of_DP arrays. matrix Lemma 8: Let U be any finite R-module which is a subset of W. If u is an array in U, then G(%)c U, where 4 8 = a(u) is the characteristic ideal of U. Proof: Note that u and its translates ,u are contained in U. Let A(%) denote an independent point set of 8. I/12 I/* Example 4: Let v’ be the array rearranged from v (Fig. Suppose the translates ,u for I PA are linearly depen- 2) by means of dent. Then there must be a nonzero polynomial f(x,y) = ~(I,J)EA(QQh,JXIy J such that f(x,y)u=(O), which implies that u E G(%!l++(x,y))). By the definition of 9X,f(x,y) must be an element of kX,which contradicts Theorem 2. Consequently the F-module G(9.l)is spanned by the translates ,u for I EA(%), the cardinality of which is equal to Then v’ has a FP parallelogram I; X I& where dim G(%). Therefore, G(Z) is a submodule of U. Q.E.D. In view of Theorem 4 and Lemma 8, we can complete the proof of Theorem 1 (seethe beginning of this section). 124 IT-24, NO. 6, NOVRMRER 1978 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. Example 5: Let q = 2. Consider %=(x4 + x2 + 1, (x2 + x + l)(y + x), y2 + x2) (= B2 in Example 3). For 0’ . . . :o;oiooo... T=II -: 2342l&‘(9q=(y-4+y-2+ iill 1,(y-z+y-‘+1) (xy2+y-‘),x2y4+y-2) =(y4+y2+1,(y2+y+l)(x+1),x2+1), where dim G(X) = dim G(B) = 6. So far we have considered only linear invertible mapFig. 4. Example of rearranged array and its FP parallelogram. pings +s corresponding to unimodular matrices S. We now introduce a generalization of rearrangement and of and per (v) = per (0’) = 12, per, (v’) = 2, per,, (v’) = 6. (See decimation as defined by Nomura et al. [2]. Fig. 4) Definition 7: For a nonsingular integral matrix Clearly the number of times that each element of F appears in any FP parallelogram I, X I, of a DP array is A=; f;, left invariant under rearrangement by means of the elementary unimodular transformations the array u=+,(u) such that v($= u(a), iE Z2, is said to be the A-decimated array of u. Lemma 9 is also valid for any decimation +A. By analogy with the ring-automorphism I/.+ we introduce the endomorphism $A for a nonsingular integral matrix A. Definition 8: For which generate the unimodular transformation group. This yields the following lemma. The weight of I, X I2 is A=; ;, defined to be the number of nonzero values on the points contained in I, x I,. E R”, qAcf(x,y)) = Lemma 11: A FP parallelogram of a DP array has the det (A)# 0, and for f(x,y) same weight as that of any equivalent array. f’(X,Y) p f(xV,xV). #4 is a-unitary injective endomorphism of l?. If $!lis an Corresponding to I, x I2 with ideal in R, so is the inverse image $i ‘(VI). Thus we have an extension of Theorem 6. II II II II Theorem 7: +A( G(X)) = G($; ‘(5%)). the set of lattice points r= {(i,j)lO<i<r, 0 < j<s”} or {(iJ)lO<i<r’, 0 < j<s} is said to be a unit rectangle of u. A DP array has the same distribution of elements of F in a unit rectangle as in a FP parallelogram. This implies that equivalent DP arrays corresponds to equivalent codewords. One of the aims of the following is to study the equivalence between Abelian group codes. From the linearity of rearrangement & and since ts(G(%)) is closed under translations, &(G(%)) is an R-module of DP arrays. From Th_eorem1, &( G(a)) = G(%‘) for a certain idcal as in R. Thus +s induces a ring-automorphism of R. Any automorphism of l? is of the form: 1CsCf(x,r))=f(~~~,x’r~), where s= II ; f; II Proof: Let v =$A(u) for an arbitrary array u in G(8), and consider the operation f (x,y) = Xck IJErcffk, [x 5, ’ E $;‘(X) on the array 0. We have the identity 2’kErU,fkv(i +k)=Z kErcffku((i+ k)A), which is the i’-component of the array f’(x,y)u and which must vanish identically, where i’ = i4 and f’(x,y) = #AcIAcf(x,y)) E X. Conversely, define u by u(iA +m)=v(i) for an array VE G($;i(%)), where m runs through each coset of the additive group Z2 modulo the subgroup Zi 4 {i’li’=iA, iEZ2}. Thus u is uniquely defined from v by a choice of the coset leaders m. Then for f (x,y) E 4; ‘@I), we have the identity 0= where the set c kErc,,fkv(i+k)=CkErCffk~((i+k)A+m), of points { i’li’= iA + m} coincides with Z2. Therefore, we have f’(x,y)u =(O), where f’(x,y) = IcIAcf(x,y)) E%. Thus Q.E.D. 14E Gt W, and 0 = +A(,,p> E 4, t G(W). Example 6: Consider u E G(s), ‘%=(x4 + x2 + 1, (x2 + x + l).(y + x), y2+ x2). (See Fig. 2.) For is a unimodular matrix. Theorem 6: (p,( G(X)) = G(I& ‘(5X)). Corollary 5: dim G(a)= dim G($;‘(%)), = dim h( G(%)). v’=+~(v) and u”=+~(,,~u)EG(!J~), !8 f $i1(%)=(x2+ 1, i.e., dim G(%) y2 + 1, xy +y + x + l), are shown in Fig. 5, where dim G(B)=3. 725 SAKATA: DOUBLY PERIODIC ARRAYS vx u . . . . . . . . . . . . . . ..o..... . . ..lO..... OlOO.... . . . . . ..o.... ::iooll.... .0100010... . . Ollll.... . . OOlO..... . ..ll...... . ..o....... . . * . . . . . . . . v' . . . . . . . . .OlOlO.. 01010.. :ololo.. . . . . . . . . . . . . . . . . . . . . . ..l.... . . . . ..l.O... . . . ..O.O.l.. . . ..O.l.l.l. . ..l.l.l.l.O ..l.l.l.O.O. . ..l.O.O.l.l . . ..O.l.l.l. ... ..l.l.l.. .... ..l.O ..... ..o ............ 0 1 1 0 0 0 1 0 0 I 0 v" 0 L . . . . . . .OlOlO.. . 10101.. .OlOlO.. . . . . . . . . . . . . . . . . 1 0 . 1 0 0 1 ..... ............ ..o .... ... .... 0 1 0 1 . . . . ..l.O... . . . ..l.l.l.. . . ..l.O.O.O. . ..l.l.O.O.l ..O.O.l.O.l. .0.0.1.1.1.1 ..0.1.0.0.0. . ..l.l.O.O.l . . ..O.l.O.l. . . . ..l.l.l.. . . . . ..o.o... . . . . . . . . . . . . 0 1 0 1 . . . . . . ..o.... . . . . ..llO... . . . . . 11101.. . . . . 1000010. . . . 101101011 0101110110 :01011110101 . . 0110000101 . . . 101101011 . . . . 0111011. # . . . . 11101.. . . . . ..ooo... . . . . . ..o.... . . . . . ...*... Fig. 6. Example of interleaved array. 0 1 0 Fig. 5. Examples of decimated arrays. V. IRREDUCIBLE LINEARRECURRINGARRAYSAND M-ARRAYS Clearly, we have +A(G(a) + G(B)) = G~((G(%))+ (pA(G@)). In the above proof, we have introduced an operation which is dual to decimation Go. Generalizing it, we define two-dimensional interleaving. Definition 9: Tp arrays in the set +:(G(%)) 4 { u]+~ ~~G’Q ig’z > are called A-interleaved arrays of Since a zero-dimensional prime ideal !@is maximal in R”, G(p) is a minimal g-module of DP arrays (except for the trivial module ((0))). An arbitrary nonzero array in a minimal $-module G(Q) is said to be irreducible. The algebraic set of @ consists of algebraically conjugate zeros over P=GF (4): I’(!@= {(a,P),(a!q,Pq),. * *, +g7- ,pqK-‘)}, where a,/? are nonzero elements of an Provided S is a unimodular matrix, +i = +; ’ = +s- ]. In extension field GF (qL) of F, and KI L. Thus cr,p can be general, as operations on sets of arrays expressedas powers of a primitive element 0 of GF (qL): (~=f?~, /3=Bx. Then K=(F’(Q)(=min {K>O(W~-~ +,4*+:=1, (7) mod N}, where p g gcd (K,X) and N f qL- 1. !l3is generbut rp:~*.+~ # 1. The following theorem is proved in Appenated by irreducible polynomials: !l3= (J(x), h(x,y)) = dix D. (g(y), h(x,y)). K=lcm (K,,K,), where K, =deg f, the deTheorem 8: +z(G(%)) = G(XJ, where sI, is the mini- gree of f(x), and K, = deg g. The degree in 2 of h(x,y) is mum ideal that satisfies #; I(%,) =% and contains GA(%). deg, h = K; = K/K,, and the degreein x of h(x,y) is deg, Furthermore, dim G(aA) is equal to dim G(a) multiplied h = K; -‘K/K,. Both {(i,j)lO <i <K,, 0 <j <K;} and by the absolute value of det (A). {(i,j)lO<i<K{, OCj<K,} are independent point sets. An arbitrary array u in G(p) is given as follows: Example 7: Consider k!I= (x4 + x2 + 1, (x2 + x + l)(y + x), y* + x2) in Example 6. For UiJ= ,,zK-, ~~(&9~*, (id EZ* (8) Y&=(x4+1, (x*+l)y*+(x+x3)y+x*+l, y4+y2+(x+ x3)u + x2), where dim G(%,) = 12= 6.2, det (B) = 2 and dim G(a) = 6. An interleaved array c* E G(%,) is shown in Fig. 6. A translate of &(u*) yields 2)E G(%) in Figs. 2 and 5. For !B in Example 6, !X is not the minimum ideal containing #A(!@, but BA =(x6+ 1, (x3+ 1)~+x4+x,y2+ x2) is. dim G(BA) = 9 = 3.3, and the interleaved arrays of G(B) are G(%,& but not G(a). where yk, 0 <k < K- 1, are elements of GF (q ‘). More specifically, let L = K. Then a nonzero array is given as uiJ= ,,k:K-, t3(“+rri+Aj)qk, (ij)EZ* (9) where M is an integer such that 0 < M < q K - 2. Thus, as is easily shown, the partial periods of any nonzero array u in G(\SB)are obtained as per, (u) = r % min { r > 01rK c 0 mod N} = N/gcd (N,K), perv (u)=s p N/ gcd (N,A), where 726 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-%, N f qK- 1. As for the period, we have the following result. Theorem 9: G(Q)--{(O)} consists of (qK - 1)/t cycles of length t P lcm(r,s)=min {t>O(tp=OmodqKl} =(qK- l)/ gcd (qK- l,d. A FP parallelogram I, X f, of any nonzero array in G(q) is given by NO. 6, NOVEMBER 1978 iii) there is a pair of nonsingular matrices A and A’ such that +A(G(p)) = G(!@‘)and +AA’( G(V)) = G(%). Example 8: Let q=3. Consider the following irreducible polynomials over GF(3): f(x)=xZ+x+2, g(y)=y4+2y3+y2+2y+ 1 ~(y2+xy+l)(y2+2(x+l)y+1)modf(x). For a root B of x4 + x + 2, which is a primitive element of GF (34), 0*O= 1, f(x) has a root (Y= 0 lo, (;YEGF (3*). The where r’ = t/s, sn = t/r and s’= ~(1- m//m), rN = r(lprime ideal ‘Y =(x2 + x + 2,y* + xy + 1) has the algebraic m”/m) for m 4 gcd (r,s)=rs/t and certain integers set ~(q3)={(e~~,e*), (e30,e24), (e10,e7*), (030,056)}, and m’,m”,0<m’,m”<m.5 perx ($) = 8, perv (9) = 10, per (!$3)= 40. For We now consider the rearrangement or decimation +A of irreducible arrays, where A=; II ;, II Q’ 2 ~i’(~)=&m3)=( x4+x3+x*+l,y+x3+2x*+x), perx ($Y)=40, perY (@‘)= 20, per ($‘) =40, and V($‘) = {(e38,e28),(e34,e4),(e**,el*), (e66,e36)j.An array u in G(q) and the rearranged array u’ = &s(u) E G(!@‘) are shown in Fig. 7, and the decimated array c =+A(u) E G(‘tJ3’)is shown in Fig. 8. Corollary 6: An arbitrary irreducible array can be rearranged to form the perpendicular repetition of a (simple) periodic sequence, that is, there is a unimodular matrix S such that per, (r#~~(u))=per(u) and per,, (&(u)) = 1. K’-aK+ bh, A’=cK+dh mod N( g qL- 1) (10) It follows from this corollary that an irreducible two-dimensional cyclic code of code area m X n is equivalent to where Zi is the direct product of the coset group Z/(qL - 1) with itself. In order to give the condition that the an irreducible cyclic code of code length mn, provided period is left invariant under a decimation +A, we need the that m, n are relatively prime to q.6 Example 9: The 8 X 10 rectangle of u in Example 8 is following lemmas. Lemma 12: For given integers K, X, N, there are in- composed of two unit rectangles. For tegers u,r such that gcd (K,X, N)=gcd (K + oN,X+ TN). Lemma 13: If gcd (K,X) = gcd (K’,A’), then there exists a unimodular matrix S such that K’ = aK + bX, x’ = CK i- dX, @‘s((10,8))=(2,0), and Q’=$J~‘($~)=(x~+x*+~x+ I,y+ where 1). An 80 x 1 rectangle of an array in G(@‘) is a codeword of the cyclic code with the check polynomial x4+ x2+2x + 1. The following definition7 is a two-dimensional generaliFor the proof of Lemma 13, we remark that p( p gcd zation of M-sequences [19], i.e., linear shift-register (K,A))= aK+ bh for certain integers a,b and that, for c 2 sequenceswith maximum length. --h/p, d p K/P, ad- bc = 1. The following theorem is Definition 10: If a DP array u has per (u)= qK- 1 for proved in Appendix E using these lemmas. K A dim G@(u)), then it is said to be a maximum period Theorem IO: Two irreducible arrays u E G(p) and o E array or simply an M-array. G(@‘) have the same period per (u) = per (v) if and only if The above definition is invariant under rearrangement any of the following conditions is valid: by any unimodular transformation. If % is the characteris{(O)} consists of tic ideal a(u) of an M-array u, G(a)i) there is a unimodular matrix S such that &(G@)) qK1 translates of u which are all M-arrays with period = G(W); qK1. ii) there is a nonsingular matrix A such that gcd(det The following theorem shows that an M-array is a (A>,#‘- l>= 1 and +AGG9)= GW); two-dimensional analog of an M-sequence. det (A)#O. For a prime ideal @ belonging to the algebraic set IQ) which contains (cr,/3), let the corresponding prime ideal $3’= $i i(p) belong to the algebraic set I’(@‘). Then V(Q’) contains (a’,/?‘), where a’= aapb, p’= aCpd, and K’= dim G(!$‘) is less than or equal to K=dim G(q). Let a=er, ,f3=@‘, a’=eK’, /3’=@” for a fixed primitive element 19of GF (qL). Then a rearrangement or decimation +A induces a linear transformation OA from Z,,$into itself: s=; 5;. II /I 51n order to determine the integers m',m", we remark that there exist positive integers a’, A’ relatively prime to N such that KGK’ gcd (K, N), ptime to m, there Ad gc4 (A,!) mod N [19! A S K’, h’ ue EdSo r&tiVdy exist posttive integers K’- ,x’-’ such that K’K’-‘d’X’-*E 1 mod m. Then m'~:h'-lK',m"~K'-'h'~m'-' mod m. ‘In case of gcd (m,n)# 1, the result has been unnoticed as far as the author knows. ‘This definition differs from that of Nomura er al. [l], [2], where the dimension is ignored and the concept of “period” is different from ours. 727 SAKATA: DOUBLY PERIODIC ARRAYS u . . . . . . . . . . 1 . . . . . . . . :01....... .llZ...... .2001..... .212111... .021n10... .221101... 100212... :121222... . . . 2020... . ...202... . . ...21... .... ..l... .......... -, u’ . . . . . . . . . . .112122... .(I 12 2 11 is \ il . 3 N & qK- 1, N’ &i qKl-- 1, let a=e(N/N’)o’, &‘I N’, if N’/u’]qk-- ii, . 0 2 1 11 .?12...... .z........ . . . . . . . . . . 1. (13) (14) Theorem 12: An M-array is a yp-array, and vice versa. 2 2 2 0 2 . .. . . . . . .. 1, then k > K,, (12) Nomura et al. [2] define a y&array to be what in our terminology is an array in G(p), where !@is the prime ideal with the algebraic set V(Q) containing a zero (a,p) satisfying (12) (13) and (14). $-arrays are easily shown to be M-arrays. The following theorem shows thtit conversely all M-arrays are y/?-arrays. (SeeAppendix F.) 1 . . where , gcd (+,(N/N’)a’)= 1 .llLl..... . 2 20 0 . . . . . p=eqfl, u’, 7’, n are positive integers such that gcd (7, N) = 1, . . . .120112... . (I 2 0 0 1 . ....... ....... ...... ...... ....... ....... . . .... ....... ...... ...... ...... ....... ..... ....... plies that an M-array is rearranged to an M-array by means of any unimodular transformation. For a primitive element 0 of GF (qK), K= K, K;, From Corollary 6 we can obtain a generalization of the one-to-one correspondence between a yp-array and an M-sequence described in T. Nomura et al. [2, th. 91. 0. 2 . 1 . Fig. 7. Example of irreducible LR array over GF (3) and its rearrangement. Theorem 13: An arbitrary M-array can be rearranged to the perpendicular repetition of an M-sequence with the same period. The construction of pseudorandom arrays from pseudorandom sequencesshown by F. J. MacWilliams and N. J. A. Sloane [4] is also a special case of Theorem 13. Example 10: Letq=2*,GF(2*)={0,y,y2,1(=y3)}.For a primitive element 0 of GF (22’3)such that 0 3+ B* + ye + y2=e6+e+i= 0, the’ prime ideal !Q= (x3 + yx* + 1, y + yx*+x) has the algebraic set V(Q)= {(015,r5i’“),(060,840), (85*,f334)).Thus perx @3)=21, perY@)=per (!@)=63. For v . . . . . . . . . . .1022121.. .1002112.. .2221022.. .lOOlO.... .21222.... .2211..... .0112..... .oo....... . . . . . . . . . . f+ Fig. 8. Example of decimated irreducible array over GF (3). II -2 1 -1 II 3’ qJ’=l+g’(q3)=(x3+ yx* + y*x + y, y + I), per (!$Y)= perx (Q’) = 63, per,, (!@)= 1. A unit rectangle of an M-array u E G(Q) corresponds to a period of an M-sequence 2)E G(@‘) with V, = uij, where the correspondence between k From Definition 10, it follows that the characteristic and (ij) is shown in Table I. ideal of an M-array is maximal in i. Thus we have the Theorem 13 also shows that the pseudorandomnessof following lemma. an M-array is relevant to that of the corresponding MLemma 14: The characteristic ideal % of an arbitrary sequence. As a pseudorandom (rectangular) array, we M-array u is prime in i, i.e., u is an irreducible array. may make use of a unit rectangle of an M-array u instead From this lemma, the earlier results given in this section of a FP parallelogram of u. Let p,(z) denote the autocorrecan be also applied to M-arrays. lation function of a DP array u, defined by Lemma 15: Let K= dim G@(u)) and 8 be a primitive element of GF (qK). Provided that a zero (0’,0”) is i=(i,j)EZ* contained in V@(u)), u is an M-array if and only if p,(i) 22 E 17Mj)) 77Wfj)) T Theorem 11: For any integral vectors i,j and arbitrary elements c, d of F, c&) + d($) is either (0) or a translate of u if and only if u is an M-array. jEr gcd (K,&qK- I)= 1. (11) Condition (11) does not depend on the choice of a primitive element of GF (qK) and is invariant under unimodular transformations (K’,xl) = QS((~, A)), which again im- where I is a unit rectangle of the array u and n is the isomorphism from the additive group of F=GF (q) to the multiplicative group of complex qth roots of one. Invoking Zierler [19, corollary to th. 121,we have, in view of Theorem 13, the following result. 728 IEEETRANSACTIONSON INFORMATIONTHEOR~,VOL. IT-24,NO.6,NOvmmm VI. TABLE I CONSTRUCTIONOFAUNITRE~TANGLEOFANM-ARR~YPROMAN M-SEQUENCE k j0 1 2 0 2 4 5 7 6 8 10 i 0 13 2 3 9 11 13 4 12 14 16 19 5 15 17 6 18 20 22 7 21 23 25 8 24 26 28 9 27 29 31 10 30 32 34 11 33 35 37 12 36 38 40 13 39 41 43 14 42 44 46 15 45 47 49 16 48 50 52 17 51 53 55 18 54 56 58 19 57 59 61 20 60 62 1 Theorem 14: Let t=(qK-1)/(4-l). An M-array u with period qK- 1 has the following autocorrelation function: i) for i=(i,j) such that ai+cj&k mod qK-1, O<k< qK- 17dl?= 4K-2~A,,EFm~( d - ls(O>129 ii) fori=(i,j)such thatai+cjrtkmodqK-l,O<k< qK- 19p(r?=4K-1~hEF~l(~)71(~kX)-l$(0)12, where s= ; II 5; II is the unimodular matrix corresponding to the rearrangement from the perpendicular repetition of an M-sequence to u and Y is a primitive element of F. In particular, for q = 2, we have the following corollary. Corollary 7: An M-array over GF (2) with period t = 2K - 1 has the autocorrelation function p(i)= - 1, t 9 i i#nd, + nl, i=mI,+n12 where (m, n) E 2* and I, X 1, is a FP parallelogram of u. When applied to M-arrays, Theorem 10 yields the following theorem. Theorem 1.5: Two M-arrays u,v have the same period tzqK- 1 if and only if there exists a unimodular transformation S such that v =+&) for certain 1E Z*. 1978 CONCLUSION We have given a general treatment of doubly periodic arrays over a finite field GF (q). Linear recurring arrays over GF (q) are shown to coincide with DP arrays. A DP array is determined by a system of LR relations, given the values in an independent point set. Any independent point set associated with the system of LR relations can be obtained in general. Those properties which are invariant under rearrangements of arrays by unimodular transformations are the most important in studying DP arrays. In particular, rearrangement, decimation, and interleaving determine correspondencesbetween arrays. The general treatment of DP arrays is applied to irreducible arrays. M-arrays, which are a special class of irreducible arrays, are dealt with from the same point of view. A two-dimensionally proper definition of M-array is given. Correspondences between irreducible arrays are clarified by means of rearrangements and decimations. The general construction of M-arrays from M-sequences is given by unimodular transformations. M-arrays are shown to coincide with -&arrays. Thus it becomes clear that there exist no arrays having “maximum period” in our terminology other than yp-arrays. DP arrays are a natural generalization of linear recurring sequences[19]. Using our results, we can implement LR arrays and their transformations by a two-dimensional feedback shift register. It is also easy to extend our investigation to multidimensional arrays. ACKNOWLEDGMENT The author wishes to express his sincere thanks to Prof. M. Iri of the University of Tokyo for his valuable advice. The idea of “rearrangement” was suggestedby him. Particular thanks are due to Dr. T. S. Han of the Sagami Institute of Technology, who read parts of the manuscript and contributed many helpful discussions. The author is also indebted to Dr. H. Imai of Yokohama National University for his very helpful comments. APPENDIX A “THE RING I? OF OPERATORS IS NOETHERIAN” For an ideal % in l?, % g %n R is an ideal in the polynomial ring R. Conversely, the ideal % in l? is generated by the elements of %. Thus an ideal in l? is in one-to-one correspondence with an ideal in R, from which it follows that the divisor chain condition [ 171is valid in a as well as in R. This means that k is also a Noetherian ring and almost all theorems about ideals in R can be restated for ideals in R”. Every ideal in I? has a finite basis (j(l), . . . JCL)), where f(l), . . . JCL) may be chosen to be polynomials. Furthermore, the Lasker-Noether decomposition theorem is also valid [ 171, [ 181: every ideal ‘% has an it-redundant representation as the intersection of a finite number of greatest primary components, %= n $Li. These primary ideals Qli belong to distinct prime ideals !J.$. SAKATA:DOUBLYPERIODICARRAYS 729 APPENDIX B SOME THEOREMSON R-ISOMORPHISMS [ 18, PP. 132- 1441 APPENDIX E PROOF OF THEOREM 10 Let R be a commutative ring. For submodules N and L of an R-module M with N c L, the difference module L-N (or the factor module L/N) is a submodule of M - N, and (M- N) (L- N)=ML, where MzM’ indicates that M and M’ are R-isomorphic. If N and L are submodules of an R-module M, then (L + N) - N = L - (L n N), where L + N is the “sum” of L, N. Let I be an ideal in a ring R. The residue class ring R/9X and the difference module R -‘?I are identical as sets, as additive groups (modules), and as R/%-modules. If R contains the finite field F, an R-module is also an F-module. A finite R-module M has finite dimension dimF (M) as an F-module. From the above isomorphism theorems, we have the following lemmas. Lemma BI: For ideals N, % in R with N ~8, dim, (R/N)dim, @I/N) = dim, (R/Q. Lemma B2: For ideals %, b in R, dim F (%+B/B)=dim, Condition i) is a special case of ii), and ii) is a special case of iii), because in the case of ii) there exists an inverse map @il. Under condition iii), there are integers e,f such that K’ = aK + bh + e(qL - l), h’= CK+ dh+f(qLl), from which it follows that gcd (K,A,qL.- l)j~‘,h’. Then we have gcd (K,X,qL- l)]gcd (K’,x,q’1). Similarly, gcd (K’,x,qL1)lgcd (K,A,qL- l), so that we have gcd (K, A, qL - 1) = gcd (K’,X, qL - 1). Consequently per (u) = per (u). Conversely, assuming gcd(K, A,qL - 1) = gcd (K’,x,q‘1), we have, by virtue of Lemma 12, gcd (~+u(q~1), X+r(qL-l))=gcd (d+o’(qL-I), x’+T’(qL-1)) for certain integers u, T, (I’, 7’. Therefore, in view of Lemma 13, there exists a unimodular matrix (%/an 8). s= such that ; I/ ; II Q.E.D. (K’,~)=@s((K,~)). APPENDIX F PROOF OF THEOREM 12 APPENDIX C DETERMINATION OF THE CHARACTERISTICIDEAL a(u) We begin by finding an independent point set A 2 A@(u)) of the characteristic ideal a(u) of u. For any finite subset l? of Z2, theperiphery Xof I’is definedas: ar={r+i,lrEr}u{I+~IIE T} -T. Let I be a FP parallelogram (or a unit rectangle) of u l?k+l gY,+{m,}, where rn,E and T,=(O). For k=1,2;.., X, is chosen so that, for a certain iET and for arbitrary4 E F, jerk, u~+,,,,+X~~~&~+~#O. If there exists no such m,EX, for some k, then A=Tk and k =dim G@(u)). For each l=(k,I)~aA, we find a polynomialf,(x,y)=x~‘+ Z~i,jjE&x$j satisfying j&=(O). Then X=({fr]l EVA}) is the characteristic ideal g(u). APPENDIX D PROOF OF THEOREM 8 From (7), +2(G(YI)) =+i ‘(G(X)). In view of the linearity of +“, for u, u~+;‘(G(?l)) and c,dEF, we have +A(cu+du)= c+~(u)+~+~(o)E G(a), from which it follows that cu+ du E +i ‘(G(a)). From the definition of $3, xu E+~(G(%)), provided that u E+$(G(%)). Let r = per, @I)),s = perY (X). Then for u E +2(G(%)), we have u.r+ark+Esl,jj+brk+ll=ui,jr (idi), (k,4EZ2. In particular, Ui+rsfO&&j,j = uij = Uij+,s(a&bc), (i,j) E z2. Thus +;A*( G(%)) is an R-module of DP arrays, which implies, in view of Theorem 1, that there is an ideal %A in I? such that +;(G(lu))= G(91u,).Since +“(G(YXA))= G($il(YIIA))= G(g), +il(YIA)=% by virtue of Lemma 6. Furthermore, an arbitrary array UE +J(G(%)) satisfies, for f(x,y)E%, ZkErUrfku((k+i)A +m)= X:kerCf2fku(k+i)=0, where u=+~(,,,u)EG(%), i,rn~Z~. Therefore, ZkerU).fku(k,4 +i)=O, iE Z2, from which it follows that f(x,y)u = (0), where f’(x,y) = #,&(x,u)) E #,&I). Accordingly aA contains lclA(91).Suppose B be any ideal containing $“(‘%). Then $7 l(B) I#; ‘(4” (a)) = %+ ker $A = ?I = 42 ‘(gA), where the kernel ker $A of the one-to-one map 4” is the null ideal (0). Therefore, B 1 aA. The final identity follows from the fact that ]Z2/ ZjI is equal to the absolute value of det (A) and that there is a unique array u corresponding to choosing the arrays in G(‘%) one by one from Q.E.D. each coset. Let N’=q Kl - 1, N d = qK2 - 1. For a primitive element 8, = eNIN’ of GF(qKI) and a primitive element e2= BNINV of GF (qG), (Y=B%,/3=0” can be rewritten as LY=&‘, R=&. Therefore, K = (N/ N’)u, X = (N/ N”)r. Condition (I 1) implies that gcd (N/N’,N/N”)= 1, from which it follows that N=Icm (N’,N”). This yields N<N’N”<q K~+K2- 1 and thus K<K,+ Kp Then, assuming that K2 > K,, K2 > K, >K- K2 = K,(K; - l), and K; = 1, i.e., K= K2. In this case X=r. Let u=&’ mod N’, r=lr’ mod N, where (I’ k gcd (a, N’), r’ k gcd (7, N), and gcd (5;N’) = gcd (s,N)= i 1191.e; k ef 1s a primitive element of GF (qKI) and 0; 2 e5 is a primitive element of GF (qK). There exists a primitive element 0’ of GF (qK) such that 0; = 0’N/N’, 0; = V, for a certain integer n relatively prime to N. Thus, writing B instead of 8’, we can put K = u’(N/N’), h= nr’. Condition (13) is clear. Condition (11) implies that gcd (u’,r’, N) (= gcd (u’, 7’)) = 1 and gcd (N/N’,+) = 1, from which it follows that 1= gcd (N/N’,X,N)=gcd (A, N/N’), i.e., r’=gcd (&N’)(N’. Therefore u’r’]N’ (12). gcd (r’,u’(N/N’))= 1 (14) has already been established. Q.E.D. REFERENCES 111 T. Nomura and A. Fukuda, “Linear recurring planesand two-dimensionalcyclic codes”(in Japanese),ZECE Trans., vol. 54-A, pp. 147-154,Mar. 1971. PI T. Nomura, H. Miyakawa, H. Imai, and A. Fukuda, “A theory of two-dimensional linear recurring arrays,” IEEE Trms. Inform. Theory, vol. IT-18, pp. 775-785, Nov. 1972. 131 T. Ikai, H. Kosako, and Y. Kojima, “Two-dimensional linear recurring arrays--Their formulation by ideals and eventually periodic arrays,” ZECE Trans., vol. 60-A, pp. 123-130, Feb. 1977. (41 F. J. MacWilliams and N. J. A. Sloane, “Pseudo-random sequencesand arrays,” Proc. IEEE, vol. 64, pp. 1715-1729, 1976. [51 S. Sakata, “Multi-fold linear recurring arrays and multi-fold Marrays” (in Japanese), Technical Group on Automata and Languages, IECE, Japan, AL76-66, Dec. 1976. “Multi-fold linear recurring arrays and their rearrange[61 -, ments” (in Japanese), Technical Group on Automata and Languages, JECE,.Japan; AL76-84, Mar. 1977. “Doubly linear recurring arrays and M-arrays,” ZECE [71 -, Truns., vol. 60-A, pp. 918-925, Oct. 1977. 181H. Hancock, Lectures on the Theory of E&tic Functions, vol. 1, Anu&sis. New York: Wiley, 1910. 730 IEEETRANSACTIONSON INFORMAT~ONTHEORY,VOL.IT-%,NO. 6,NOvu~umt 1978 191 S. D. Berman, “Semi-simple cyclic and Abelian codes,” Cybernetics, vol. 3, no. 1, pp. 21-30, 1967. 1101 P. Delsarte, “Automornhisms of Abelian codes,” Philips Res. Rep., vol. 25, pp. 389-403, i970. .I1 11 , F. J. MacWilliams. “Binarv codes which are ideals in the aroun algebra of an Abdlian group,” Bell Syst. Tech. J., vol. 4% pp. 987-1011, 1970. [12] H. Imal, “Two-dimensional fire codes,” IEEE Trans. Inform. Theory, vol. IT-19, pp. 796-806, Nov. 1973. [ 131 M. Arakaki and H. Imai, “Theory of two-dimensional cyclic codes of even area” (in Japanese), Technical group on Automata and Languages, IECE, Japan, AL73-77, Feb. 1974. [ 14) T. Ikai, H. Kosako, and Y. Kojima, “Basic theory of two-dimensional cyclic codes-Periods of ideals and fundamental theorems,” Generalized Quadratic JACOBUS H. VAN LINT AND Abstract-A simple definition of generalized quadratic residue wdes, that is, quadratfc residue codes of block length p m, is given, and an account of many of their properties is presented. 1. ZECE Trans., vol. 59-A, pp. 216-223, Mar. 1976. [ 151 -, “Basic theory of two-dimensional cyclic codes-Structure of cyclic codes and their dual codes,” ZECE Trans., vol. 59-A, pp. 224-231, Mar. 1976. (161 -, “Basic theory of two-dimensional cyclic codes-Generator polynomials and the positions of check symbols,” ZECE Trans., vol. 59-A, pp. 311-318, Apr. 1976. [ 171 B. L. van der Waerden, Modern Algebra, vol. II (F. Blum, English transl.). New York: Ungar, 1949. [ 181 0. Zariski and P. Samuel, Commufatiue Algebra, vol. I. Toronto: van Nostrand, 1958. [19] N. Zierler, “Linear recurring sequences,” J. Sot. Zndust. Appl. Math., vol. 7, pp. 31-48, 1959. INTRODUCTION ET p,l be distinct primes such that 1 is a quadratic residue of p. Let IF= GF(Z), and let G be the Abelian group of the additive structure of GF(p). The (classical) quadratic residue codes A +, B +,A, B are certain ideals in the group algebra [FG. G is of course a cyclic group, and the group operation is written as multiplication, i.e., G= {1,x,x2; * * ,xp-1). A + is defined as follows. Let .$ be a primitive pth root of unity over IF; a polynomial c(x) of 5G is in A+ if c([‘) = 0 for all r which are quadratic residues of p. The code A has one as an additional zero. B +, B are defined similarly with respect to the nonresidues of p. (See [l], [6], [8].) We wish to extend this idea to codes of block length 4=pm, m > 1. These will be called generalized quadratic residue (GQR) codes. The restrictions on 5 will be described later; in fact, if m is even, there are no restrictions except that IFshould not have characteristicp. ff can even be taken to be the real numbers. G is now the Abelian group of the additive structure of GF(p”). It is no longer cycliclmbrt is an elementary Abelian group of type , p}, which means that it is the direct product of {p,p,*** m cyclic groups of order p. The code positions will be Residue Codes F. JESSIE MAcWILLIAMS identified with the elements of G. Since we need addition and multiplication both in GF(pm) and in the group algebra IFG, we use the symbols @ and * for these operations in IFG; a sum in IFG will be denoted by B. We remind the reader that IFG consists of all formal sums B ,eca,g, ugE IF, with the following rules for addition and multiplication: L (1) A subset S of G can be interpreted as an element of IFG by taking ag=l if gES and ag=O if g4S. We use the same symbol for the set and the corresponding element of IFG. In particular, U, V,O are the elements of [FG corresponding, respectively, to the set of nonzero squares of GF(q), the set of nonsquares, and the single element (0). A vector c with coordinates cg is identified with the element E, Eccgg of IFG. We denote the usual inner product of two vectors c, c’ by (ccc’). A character 4 of G is a homormorphism of G into the set of pth roots of unity over IF.The characters of G form a group x which is isomorphic in many ways to G, let us say $++g. The exact form of the isomorphism we need will be discussed in Section II. For a detailed account of the properties of characters as applied to coding theory, see [7]. A character is extended in the obvious way to a Manuscript received February 9, 1978. J. H. van Lint was with Bell Laboratories, Murray Hill, NJ, on leave linear functional on the group algebra from the Department of Mathematics, Technological University Eindhoven, Eindhoven. The Netherlands. F. J. M&Williams is with Bell Laboratories, Murray Hill, NJ 07974. 0018-9448/78/l lOO-0730$00.7501978 IEEE