ON THE EXTENDED LEE WEIGHTS MODULO 2e OF LINEAR CODES OVER Z2s BAHATTİN YILDIZ AND ZEYNEP ÖDEMİŞ ÖZGER Abstract. In this work, linear codes over Z2s are considered together with the extended Lee weight that is defined as  x if x ≤ 2s−1 , wL (x) = 2s − x if x > 2s−1 . The ideas used by Wilson and Yıldız are employed to obtain divisibility properties for sums involving binomial coefficients and the extended Lee weight. These results are then used to find bounds on the power of 2 that divides the number of codewords whose Lee weights fall in the same congruence class modulo 2e . Comparisons are made with the results for the trivial code and the results for the homogeneous weight. 1. Introduction There has been a burst of activity on linear codes over rings in recent years. Starting from the ring Z4 with [1], the research in this area has been generalized to Z2k , Zpk , Galois rings and finite chain rings in general. The weight function used in codes over Z4 is the so called Lee weight that has values wL (0) = 0, wL (1) = wL (3) = 1 and wL (2) = 2. When generalizing to Galois rings and finite chain rings in general a new weight called the homogeneous weight was introduced as an extension of the Lee weight on Z4 . We refer the reader to [2] for more details. The homogeneous weight uses the algebraic structure of the rings and has few non-zero weights. It also has connections with exponential sums and thus, results from number theory can be used to obtain bounds for this weight([3], [4]). Recently, in [5], when constructing the Gray map for the homogeneous weight over Galois rings, it was discovered that the homogenous weight also has connections with hyperplanes in combinatorial Date: November 07, 2012. 2000 Mathematics Subject Classification. Primary 94B05; Secondary 94B65. Key words and phrases. Linear codes, enumerators, Gray map, codes over rings. This paper is in final form and no version of it will be submitted for publication elsewhere. 1 2 BAHATTİN YILDIZ AND ZEYNEP ÖDEMİŞ ÖZGER geometries. The algebraic constructions for the Gray map of homogeneous weight are given in [2] and [6]. However there is a more natural generalization of the Lee weight from Z4 to Z2k . This was given first in [7] by Carlet, however he chose the homogeneous weight instead of this weight to work with. Recently, Dougherty and Fernández-Córdoba have studied linear codes over Z2k with respect to this extended Lee weight in [8]. The constraints on the weight enumerators of codes have been of interest in coding theory. Pless looked at the Hamming weight enumerators of binary codes in [9]. Wilson used polynomial ideas to obtain such constraints for general weight functions on codes in a more general context in [10]. Yıldız used ideas from [10] to obtain the best possible results for the Lee weights of linear codes over Z4 in [6]. In [11] a coset decomposition idea was used to obtain the tightest bounds for the homogeneous weight enumerators of linear codes over Galois rings. In this work we study linear codes over Z2s with the extended Lee weight to prove divisibility results for the weight enumerators using ideas from [10]. Our aim is to combine tools from [10] and [6] to find the power of 2 that divides the coefficients of the extended Lee weight enumerators of linear codes over Z2s . The rest of the paper is organized as follows. In section 2, we give the necessary definitions for codes over Z2s and the extended Lee weight. In section 3, we state and prove the main lemmas to be used in obtaining the main results. Section 4 includes the main results for free Z2s -codes. In section 5 we find the results for the trivial code for comparison. Section 6 concludes the paper with remarks and possible research directions. 2. Linear codes over Z2s Definition 1. A linear code over a ring R of length n is an R-submodule of Rn ; more specifically a linear code over Z2s of length j is a submodule of Zj2s . It can be shown, because of the ideal structure of Z2s , that a linear code over Z2s of length j is permutationally equivalent to a code that has a generating matrix of the following form:   · · · As−1 As Ik1 A1  0 2Ik2 2B1 ·  · · ·   2  0  0 2 I k3 · · · ·    ·  · · · · · ·   s−2 s−2 s−2  · · · · 2 Iks−1 2 D 2 E  0 0 · 0 0 2s−1 Iks 2s−2 F ON THE EXTENDED LEE WEIGHTS MODULO 2e OF LINEAR CODES OVER Z2s 3 where A1 , · · ·, As , B1 , · · ·, D, E, F are matrices over Z2s (see [14]). As seen from the generating matrix, a dimension can not be defined for such codes. Instead of dimension, a code over Z2s has a type. A code that has the above matrix as a generating matrix is said to be of type (2s )k1 (2s−1 )k2 · · · (2)ks , and it has size |C| = 2sk1 +(s−1)k2 +···+ks . For the rest of this work we will let the extended Lee weight on Z2s be defined as follows:  x if x ≤ 2s−1 , wL (x) := s 2 − x if x > 2s−1 , s−1 for this weight function can be given The Gray map from Z2s to F22 simply as: 0 → (000 · · · 000), 1 → (100 · · · 000), 2 → (110 · · · 000), · · 2s−1 → (111 · · · 111), 2s−1 + 1 → (011 · · · 111), 2s−1 + 2 → (001 · · · 111), · · 2s − 2 → (000 · · · 011), 2s − 1 → (000 · · · 001), which is introduced in [16] for Z2k -linear codes by Borges et al. Simply put 1’ s in the first x coordinates and 0’ s in the other coordinates for all x ≤ 2s−1 , and if x > 2s−1 then the Gray map takes x to 1 + ϕL (2s−1 − x), where ϕL is the Gray map for wL . This map can be extended for a codeword c = (c1 , c2 , . . . , cn ) as follows: ϕL (c) = (ϕL (c1 ), ϕL (c2 ), . . . , ϕL (cn )). Note that ϕL is a (non-linear) isometry from (Z2s , Lee weight) to (F22 Hamming weight). 3. The main lemma We start with the following remarks: Remark 1. For any x ∈ Z2s , wL (x) + wL (x + 2s−1 ) = 2s−1 . s−1 , 4 BAHATTİN YILDIZ AND ZEYNEP ÖDEMİŞ ÖZGER Remark 2. [11] Let C be a linear code over Z2s of length j and type (2s )k1 (2s−1 )k2 · · · (2)ks with a1 , a2 , . . . , ak1 −→ free generators, b1 , b2 , . . . , bk2 −→ generators in 2Z2s , · · · c1 , c2 , . . . , cks −→ generators in 2s−1 Z2s , e be the linear code over 2s−1 Z2s generated by and let C 2s−1 a1 , 2s−1 a2 , . . . , 2s−1 ak1 , 2s−2 b1 , 2s−2 b2 , . . . , 2s−2 bk2 , · · · c 1 , c 2 , . . . , c ks , e is a linear subcode of C and, moreover then C e C = ∪ (c + C), c∈Zj2s where c’s are in Zj2s . The exact coset representatives can be found in [11]. For the rest of this paper SC is defined as the following: Definition 2. For C, a subset of Zj2s (in fact, C is either a subgroup or a coset of a subgroup) we define      X wL (aj ) wL (a1 ) wL (a2 ) , ··· SC (i1 , · · ·, ij ) := ij i2 i1 (a1 ,a2 ,...,aj )∈C where 0 ≤ i1 , i2 , · · ·, ij ≤ j such that i1 + i2 + · · · + ij = j. The following is the main result we would like to obtain. Theorem 1. Let C be a linear code over Z2s of type (2s )k1 (2s−1 )k2 ···(2)ks , then for any fixed i1 , i2 , . . . , ij with 0 ≤ i1 , i2 , . . . , ij ≤ j and i1 +i2 +···+ij = j, we have SC (i1 , · · ·, ij ) ≡ 0 (mod 22·[k1 +k2 +...+ks ]−j ). ON THE EXTENDED LEE WEIGHTS MODULO 2e OF LINEAR CODES OVER Z2s 5 Note that by Remark 2, to prove the above theorem, it is enough to prove the following lemma: Lemma 1. Let C be a linear code over 2s−1 Z2s of length j and dimension k. Then Sa+C (i1 , · · ·, ij ) ≡ 0 (mod 22k−j ) for any a ∈ Zj2s and i1 , · · ·, ij ∈ Z2s such that i1 + i2 + · · · + ij = j. Proof.  The proof is by induction on j. Let j = 1 and k = 1. Then C = (0), (2s−1 ) . Hence     wL (2n−1 ) wL (0) = 2s−1 ≡ 0 (mod 22.1−1 ) + SC (1) = 1 1 and if a ∈ Z2s is any coordinate, then Sa+C (1) = wL (a) + wL (a + 2s−1 ) = 2s−1 by Remark 1. Assume that the induction hypothesis is true for j − 1. It remains to show that the claim is true also for j. For the sake of simplicity, we drop (i1 , · · ·, ij ) in the notation. There are two cases: Case 1: Let i1 = i2 = · · · = ij = 1 and a ∈ Zj2s , a = (a1 , a2 , . . . , aj ). Then Sa+C = wL (a1 ) · S · · a+C 0 + wL (a1 + 2s−1 ) · S · · a+C 2s−1 , where Cℓ is the set of codewords that have ℓ as their first coordinates (ℓ = 0 · or 2s−1 ), C ℓ is the set of codewords whose first coordinates, which are ℓ, · are deleted and a is the codeword that is obtained from a by deleting its first coordinate. Now when deleting a coordinate from C , there can be two situations. Either the dimension does not change or two copies of a code with dimensions decreased by one occur. In the first case, the following holds: Sa+C = wL (a1 ) · S · · + (2s−1 − wL (a1 )) · S · = wL (a1 ) · S · · + wL (a1 ) · S · a+C 0 a+C 0 · a+C 2s−1 · a+C 2s−1 +(2s−1 − 2 · wL (a1 )) · S · ·  a+C 2s−1  = wL (a1 ) · S · · + 2 · 2s−2 − wL (a1 ) · S · a+C · a+C 2s−1 Now by induction hypothesis, S· · a+C ≡0 (mod 22k−(j−1) ), ≡0 (mod 22(k−1)−(j−1) ). and S· · a+C 2s−1 Hence   2 · 2s−2 − wL (a1 ) · S · · a+C 2s−1 ≡0 (mod 22k−j ). 6 BAHATTİN YILDIZ AND ZEYNEP ÖDEMİŞ ÖZGER This proves that Sa+C ≡ 0 (mod 22k−j ), · which means Sa+C is divisible by 22k−j . Now in the second case, C 0 = · C 2s−1 is a code of length j − 1 and dimension k − 1, then Sa+C = wL (a1 ) · S · s−1 =2 · S· · a+C 0 · a+C 0 + (2s−1 − wL (a1 )) · S · · a+C 0 . Hence, Sa+C ≡ 0 since S · · a+C 0 (mod 22k−j ), ≡ 0 (mod 22k−j−1 ). Case 2: Assume that one of ik ’ s is greater than 1, where 1 ≤ k ≤ j. Then, since i1 + i2 + · · · + ij = j must hold, at least one of them should be 0. Supposing that one of the ik ’s is r, with r ≥ 2, this implies at least r − 1 of ik ’ s are 0. Without loss of generality i1 = 0, i2 = 0, . . . , ir−1 = 0, ir = r may be assumed. Now,        X wL (a1 ) wL (a2 ) wL (ar ) wL (aj ) SC = ··· . ··· 0 0 r ij (a1 ,a2 ,···,ar ,···,aj )∈C Let C be the code that is obtained by deleting the first r coordinates of C. Then,     X X wL (ar ) wL (aj ) wL (ar+1 ) SC = . ··· · r ij ir+1 a r (ar+1 ,···,aj )∈C So, C is either of the same dimension of C or 2ℓ copies of a k −ℓ dimensional code for some ℓ ∈ N, where ℓ ≤ r. Then by induction hypothesis SC is divisible by 2ℓ+2(k−ℓ)−(j−r) = 22k+r−ℓ−j . But we know that r ≥ ℓ, so     X X wL (ar ) wL (aj ) wL (ar+1 ) ≡ 0 (mod 22k−j ), ··· · i i r j r+1 a r (ar+1 ,···,aj )∈C which means SC is divisible by 22k−j . 4. Linear codes over Z2s of type (2s )k In this section, the results in [10] will be specialized to the extended Lee weight enumerators of codes over Z2s by using the same techniques and tools in [10] that are introduced in the proof of the following lemma. ON THE EXTENDED LEE WEIGHTS MODULO 2e OF LINEAR CODES OVER Z2s 7 Lemma 2. [10] Let p be a prime, and e and m positive integers. Let f be an integer-valued function on the integers that is periodic of period pe . There exists a polynomial     x x + · · · + cd w(x) = c0 + c1 x + c2 d 2 of degree d ≤ (m(p − 1) + 1)pe−1 − 1 so that w(t) ≡ f (t) (mod pm ) for all integers t. The coefficients ci are integers and, moreover, ci ≡ 0 (mod pℓ ), whenever i ≥ (ℓ(p − 1) + 1)pe−1 . The following is the main theorem we would like to prove: Theorem 2. Suppose C is a linear code of type (2s )k over Z2s . If we denote by NC (j, 2e ) the number of codewords in C that have Lee weights congruent to j modulo 2e , then we have e NC (j, 2 ) ≡ 0 (mod 2  sk−2e−1 (s−1) 2e−1 (s−1)  ), where j = 0, 1, 2, . . . , 2e − 1. Here ⌊·⌋ denotes the floor function. Proof. For any nonnegative integers t and m, let d = (m + 1)2e−1 − 1, and let   d X x ci Wt (x) = i i=0 be a polynomial of degree ≤ d so that  1 (mod 2m ) if j ≡ t (mod 2e ), Wt (j) ≡ 0 (mod 2m ) otherwise, where ci ≡ 0 (mod 2ℓ ) for i ≥ (ℓ + 1)2e−1 . The existence of this polynomial is guaranteed by the proof of Lemma 2. Then we have e NC (j, 2 ) ≡ X a∈C Wt (wL (a)) ≡ d X cj j=0 X wL (a) a∈C j Given j, choose an integer ℓ such that (ℓ + 1)2e−1 ≤ j ≤ (ℓ + 2)2e−1 − 1. (mod 2m ). 8 BAHATTİN YILDIZ AND ZEYNEP ÖDEMİŞ ÖZGER Then cj ≡ 0 (mod 2ℓ ). Consider X wL (a) Tj : = j a∈C   X wL (a1 ) + wL (a2 ) + · · · + wL (an ) = j (a1 ,a2 ,...,an )∈C      X X wL (an ) wL (a1 ) wL (a2 ) . ··· = in i2 i1 i +i +···+i =j 1 2 n i1 ,i2 ,...,in ≥0 (a1 ,a2 ,...,an )∈C For fixed nonnegative i1 , i2 , . . . , in summing to j, at most j of indices iv are non-zero. For notational convenience we might assume it = 0 for t > j. Then   X wL (a1 )wL (a2 ) wL (an ) ··· in i2 i1 a∈C     X wL (bj ) wL (b1 ) , ··· Φ(b1 , b2 , · · ·, bj ) = ij i1 j (b1 ,b2 ,...,bj )∈Z2s where Φ(b1 , b2 , . . . , bj ) is the number of a ∈ C with first j coordinates b1 , b2 , . . . , bj in that order. The set of such a’s is either empty or is a coset of the kernel K of the group homomorphism C → Zj2s that projects onto the first j coordinates, which is of order 2t where t = sk − (sj − r). There are two cases: Case 1: Suppose the image of this projection has size 2sj−r , where r > j. Then K has order 2t where t = sk − (sj − r) > sk − (s − 1)j. Case 2: If the image of the homomorphism has size 2sj−r with r ≤ j, then we get   X wL (a1 )wL (a2 ) wL (an ) (4.1) ··· in i2 i1 a∈C      X wL (bj ) wL (b1 ) wL (b2 ) = 2sk−sj+r , ··· ij i2 i1 (b1 ,b2 ,...,bj )∈H where H is the image of the homomorphism, and as such, is a subgroup of Zj2s . Here H can be seen as a code over Z2s of length j. Then by Theorem 1      X wL (bj ) wL (b1 ) wL (b2 ) ··· ij i2 i1 (b1 ,b2 ,...,bj )∈H is divisible by 22j−j = 2j > 2j−r . Putting these in equation (4.1), we see that the sum in (4.1) is divisible by 2sk−(s−1)j . ON THE EXTENDED LEE WEIGHTS MODULO 2e OF LINEAR CODES OVER Z2s 9 This means that in both cases, which are investigated above,   X wL (a1 )wL (a2 ) wL (an ) ··· in i2 i1 a∈C sk−(s−1)j is divisible by 2 Now suppose k ≥ sk − (s − 1)j . 2(s−1) (m s + 1)2e−2 . Then we have   2(s − 1)(m + 1)2e−2 − (s − 1) 2e−1 (ℓ + 2) − 1   (s − 1) (m + 1 − ℓ − 2) 2e−1 + 1 ≥ = (m − ℓ − 1)2e−1 + 1 ≥ (m − ℓ − 1) + 1 = m − ℓ. ≥ This means that the inner sum of the equation is divisible by 2m−ℓ , and since by choice, cj is divisible by 2ℓ , so NC (j, 2e ) ≡ 0 (mod 2m ), where k ≥ 2(s−1) (m s + 1)2e−2 which means e NC (j, 2 ) ≡ 0 (mod 2  sk−2e−1 (s−1) 2e−1 (s−1) j = 0, 1, 2, . . . , 2e − 1. Finally, if we choose k < sk−2e−1 (s−1) 2e−1 (s−1) ,  ), 2(s−1) (m s + 1)2e−2 then the corresponding m value, is negative, which does not make sense in congruences. Note that everything done in this proof is still valid if a linear code C is replaced by a coset A = a + C of C. Hence the more general result follows: Theorem 3. Suppose C is a linear code of type (2s )k over Z2s and let A = a + C be a coset of C. If we denote by NA (j, 2e ) the number of codewords in A, that have Lee weights congruent to j modulo 2e , then we have (4.2) NA (j, 2e ) ≡ 0 (mod 2  sk−2e−1 (s−1) 2e−1 (s−1)  ), e j = 0, 1, 2, . . . , 2 − 1. Also note that when s = 2, (4.2) is transformed into (4.3) in the following theorem: Theorem 4. [6] Suppose C is a linear Z4 -code of type (4)k and let A = a + C be a coset of C. If we denote by NA (j, 2e ) the number of codewords in A, that have Lee weights congruent to j modulo 2e , then we have (4.3) NA (j, 2e ) ≡ 0 (mod 2 where j = 0, 1, . . . , 2e − 1. j k−2e−2 2e−2 k ), 10 BAHATTİN YILDIZ AND ZEYNEP ÖDEMİŞ ÖZGER This shows that the results in this paper and in [6] are consistent. The following theorem from [10] holds for any weight function: Theorem 5. [10] Let G be a group of order ps , p prime, let C be a subgroup of Gn = G × G × · · · × G, and let A be a coset of C in Gn . Suppose |A| = |C| = pk . Let µ be a mapping from G into integers and define for Pn a = (a1 , . . . , an ) ∈ Gn , µ(a) = i=1 µ(ai ). If k > s((m(p−1)+1)pe−1 −1), then for any integer t, the number N of solutions a ∈ A to the equation µ(a) ≡ t (mod pe ) is divisible by pm . Thus for a linear code over Z2s of type (2s )k [10] leads to NA (j, 2e ) ≡ 0 (mod 2 (4.4) j k−2e−1 2e−1 k . ), (4.2) and (4.4) can be compared by expanding (4.4) by (s − 1):     sk − 2e−1 (s − 1) (s − 1)k − 2e−1 (s − 1) ≥ . 2e−1 (s − 1) 2e−1 (s − 1) The above inequality is strict in many cases. For example, when s = 2, the power obtained in our case is more than twice the power obtained in [10]. 5. The extended Lee weight distribution of the trivial code over Z2s The trivial code was used in [11] to prove that the results obtained were best possible. Hence to get an idea about how close to the trivial code one can get, the extended Lee weight distribution of the trivial code will be found. The aim here is to get information about the divisibility of the extended Lee weight enumerators of codes of type (2s )k1 (2s−1 )k2 · · · (2)ks . The weight distribution polynomial of these codes is a good point to start. Theorem 6. Let C be the trivial code of type (2s )k over Z2s where s is any integer, i.e. C = (Z2s )k . Then the extended Lee weight distribution of this code is given by the polynomial (5.1) wZks (x) = (1 + 2x + · · · + 2x2 s−1 + x2 −1 2 s−1 )k . Proof. The proof is by induction on k. Let k = 1. Then wZ2s (x) = (1 + 2x + · · · + 2x2 s−1 −1 + x2 s−1 ). is Now assume that the extended Lee weight distribution of Z2k−1 s (x) = (1 + 2x + · · · + 2x2 wZk−1 s 2 s−1 −1 + x2 s−1 )k−1 . Denote the trivial code over Z2s of type (2s )k as Ck . Then Ck = (Ck−1 ∪ {0}) ∪ (Ck−1 ∪ {1}) ∪ · · · ∪ (Ck−1 ∪ {2s − 1}), ON THE EXTENDED LEE WEIGHTS MODULO 2e OF LINEAR CODES OVER Z2s11 where Ck−1 ∪ {i}’s are the set of codewords that is obtained by adding i as a coordinate to the trivial code of type (2s )k−1 . Then the extended Lee weight distribution of Ck is wZks (x) = xwL (0) .wZk−1 (x) 2 2s · · · s +xwL (2 −2) .wZk−1 (x) s 2 xwL (1) .wZk−1 (x) 2s · · · s xwL (2 −1) .wZk−1 (x). s + + 2 After plugging the generalized Lee weights into the equation and adding the terms with the same weights (5.1) is accomplished. The previous result is extended in a similar way for the case of trivial codes of arbitrary types as follows: Theorem 7. Let C be the trivial code of type (2s )k1 (2s−1 )k2 · · · (2s−i )ki · · · (2)ks over Z2s , i.e. C = (Z2s )k1 × (2Z2s )k2 × · · · × (2s−1 Z2s )ks . Then the extended Lee weight distribution of this code is given by the polynomial (5.2) wC (x) = (1 + 2x + 2x2 + · · · + 2x2 ·(1 + 2x2 + · · · + 2x2 i s−1 ·(1 + 2x2 + · · · + 2x2 ·(1 + x2 s−1 s−1 −2 s−1 + x2 −1 + x2 −2 i s−1 + x2 s−1 ) k1 ) k2 · · · s−1 ) ki · · · ) ks . Before proving the main result in this section a corollary of a theorem mentioned in [11] is needed. Theorem 8. [11] Suppose (1 + (p − 1)xp (ℓ−1)m )k ≡ A0 + A1 x + · · · + Ape −1 xp then e −1 e (mod xp − 1),  k − pe−(ℓ−1)m−1 min {υ p (Ai ) | i = 0, 1, . . . , p − 1} = (p − 1)pe−(ℓ−1)m−1 e  for e ≥ (ℓ − 1)m + 1, where υ p (k) denotes the highest power of a prime p that divides a non-negative integer k. For the case in this paper, i.e. when p = 2, the below corollary follows: Corollary 1. Suppose (1 + x)k ≡ A0 + A1 x + · · · + A2e −1 x2 e −1 e (mod x2 − 1), 12 BAHATTİN YILDIZ AND ZEYNEP ÖDEMİŞ ÖZGER then min {υ 2 (Ai ) | i = 0, 1, · · ·, 2e − 1} = for e ≥ 1.  k − 2e−1 2e−1  Using Theorem 7 we get the following theorem: Theorem 9. Suppose C is the trivial code of type (2s )k1 (2s−1 )k2 · · · (2)ks over Z2s . Then e NC (j, 2 ) ≡ 0 (mod 2 $ (k1 +k2 +···+ks )−2e−s 2e−s % ), where j = 0, 1, . . . , 2e − 1. Proof. The weight distribution polynomial of such type of a code is given s−1 by (5.2). Since all the components of this product contain x2 as the 2s−1 leading term, we can write each of them in terms of (1 + x) as follows, h h h i ks i i k k 2 1 2s−1 2s−1 2s−1 = (1 + x) − Qs (x) . − Q2 (x) ··· (1 + x) − Q1 (x) · (1 + x) All coefficients of (1 + x)2 s−1 s−1 − x2 − 1 are even, since  s−1  2 2| , i where i = 2, . . . , 2s−1 − 1. So coefficients of Qi (x)’s are even. Hence there exists qi (x) ∈ Z [X] for each i such that Qi (x) = 2qi (x). After replacing Qi (x)’s by 2qi (x)’s weight distribution polynomial becomes h i ks h i k1 2s−1 2s−1 · · · (1 + x) − 2qs (x) . wC (x) = (1 + x) − 2q1 (x) Now, using Binomial Theorem to expand each component, i hP
k1 k1 2s−1 (k1 −i1 ) (−2q1 (x))i1 · (5.3) wC (x) = i1 =0 i1 (1 + x) · · ·h i
P ks ks 2s−1 (ks −is ) is . (1 + x) (−2q (x)) s is =0 is After expanding (5.3), we see that a typical term in the expansionis of the form 2s−1 (k1 ++···+ks −i1 −···−is ) i1 +···+is (1 + x) 2 A(x), ON THE EXTENDED LEE WEIGHTS MODULO 2e OF LINEAR CODES OVER Z2s13 where A(x) is a polynomial with integer coefficients. By using Corollary 1, e the coefficients of (5.3) reduced modulo (x2 − 1) are all divisible by % $  2s−1 (k1 − i1 ) + · · · + 2s−1 (ks − is ) − 2e−1 + i1 + · · · + is 2e−1  s−1  2 (k1 + · · · + ks ) + (2e−1 − 2s−1 )(i1 + · · · + is ) − 2e−1 = , 2e−1     s−1 (k1 + · · · + ks ) − 2e−1+1−s 2 (k1 + · · · + ks ) − 2e−1 = ≥ 2e−1 2e−1+1−s   e−s (k1 + · · · + ks ) − 2 = . 2e−s Then NC (j, 2e ) ≡ 0 (mod 2 $ (k1 +k2 +···+ks )−2e−s 2e−s % ), e where j = 0, 1, . . . , 2 − 1. For comparison with the result in [10] given by (3) and the result given by Theorem 9, jlet k1 = k, kij = 0 for all i = 2, 3, . . . , s. Then NC (j, 2e ) k k is divisible by 2 k−2e−s 2e−s =2 2s−1 k−2e−1 2e−1 . But 2s−1 > 1 for s ≥ 1,k which j k−2e−1 means the theorem improves the result in [10], which is 2 2e−1 , quite considerably. Acknowledgement : We wish to thank the anonymous referees for their helpful remarks and comments. 6. Conclusion In this work, the extended Lee weight was considered for linear codes over Z2s together with its Gray map. The divisibility properties of Lee weight enumerators of linear codes over Z2s of type (2s )k were established in a similar way to what was done in [10] and [6]. Later, the divisibility of the extended Lee weight enumerators of the trivial code was examined. Comparing these results with Wilson’s results in [10] and the results about homogeneous weight obtained in [11], we see the results that are introduced here, improve Wilson’s results but the powers are not as high as the ones for homogeneous weight. This is natural to expect since the homogeneous weight of every coordinate in Z2s is already divisible by 2s−2 . The only non zero weights in the homogenous weight of Z2s -codes are 2s−2 and 2s−1 . However, the extended Lee weight takes on every value from 1 to 2s−1 . Possible direction for future research will be to reach the divisibility results for the generalized Lee weight enumerators of an arbitrary type of 14 BAHATTİN YILDIZ AND ZEYNEP ÖDEMİŞ ÖZGER code, as done for free type of codes, and to obtain the tightest bounds possible for codes over Z2s . References [1] A.R. Hammons, P.V. Kumar, A.R. Calderbank, N.J.A. Sloane, P. Solé, The Z4 linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory 40 (1994) 301-319. [2] I. Constantinescu, T. Heise, A metric for codes over residue class rings of integers, Problemy Peredachi Informatsii 33 (1997) 22-28. [3] P.V. Kumar, T. Helleseth, A.R. Calderbank, An upperbound for Weil exponential sums over Galois rings and applications, IEEE Trans. Inform. Theory 41 (1995) 456-468. [4] S. Ling, F. Özbudak, An improvement on the bounds of Weil exponential sums over Galois rings with some applications, IEEE Trans. Inform. Theory 50 (2004) 2529-2539. [5] B. Yıldız, A combinatorial construction of the Gray map over Galois rings, Discrete Mathematics 309(10) (2009) 3408-3412. [6] B. Yıldız, A lemma on binomial coefficients and applications to Lee weights modulo 2e of codes over Z4 , Designs, Codes and Cryptography (2011) (online). DOI: 10.1007/s10623-011-9512-2. [7] C. Carlet, Z2k -linear codes, IEEE Trans. Inform. Theory 44 (1998) 1543-1547. [8] S.T. Dougherty, C. Fernández-Córdoba, Codes over Z2k , gray map and self-dual codes, Adv. Math. Comm, Vol 5, (2011), 571–588. [9] V. Pless, Constraints on weights in binary codes, Appl. Algebr. Eng. Comm., Vol 8, (1997), 411–414. [10] R.M. Wilson, A lemma on polynomials modulo pm and applications to Coding Theory, Discrete Mathematics 306 (2006) 3154-3165. [11] B. Yıldız, Weights modulo pe of linear codes over rings, Designs, Codes and Cryptography 43 (2007) 147-165. [12] S. Ling, J.T. Blackford, Zpk+1 -linear codes, IEEE Trans. Inform. Theory 48 (2002) 2592-2605. [13] M. Greferath, S.E. Schmidt, Gray isometries for finite chain rings and nonlinear ternary (36, 312 , 15) code, IEEE Trans. Inform. Theory 45 (1999) 2522-2524. [14] W.C. Huffman, Decompositions and extremal Type II codes over Z4 , IEEE Trans. Inform. Theory 44 (1998) 800-809. [15] B. Yıldız, Z. Ödemiş Özger, Linear codes over Z2s with the extended Lee weight, AIP Conf. Proc. 1389 (2011) 621-624. DOI:10.1063/1.3636807. [16] J.Borges, C.Fernández and J. Rifà, Propelinear structure ofZ2k -linear codes, Technical Report Arxiv:0907.5287,2009. (B. Yıldız) Department of Mathematics, Fatİh University 34500 İstanbul, Turkey E-mail address, B. Yıldız: byildiz@fatih.edu.tr (Z. Ödemiş Özger) Department of Engineering Sciences, İzmir Kâtİp Çelebİ University, 35620 İzmir, Turkey E-mail address, Z. Ödemiş Özger: zeynep.odemis.ozger@ikc.edu.tr