On (1−u) -cyclic codes over

Applied Mathematics Letters 21 (2008) 1129–1133 www.elsevier.com/locate/aml On (1 − u)-cyclic codes over F pk + u F pk Maria Carmen V. Amarra ∗ , Fidel R. Nemenzo Institute of Mathematics, University of the Philippines, Diliman, Quezon City, Philippines Received 27 July 2007; accepted 27 July 2007 Abstract We extend the results of [J.F. Qian, L.N. Zhang, S.X. Zhu, (1 + u)-constacyclic and cyclic codes over F2 + uF2 , Appl. Math. Lett. 19 (2006) 820–823. [3]] to codes over the commutative ring R = F pk + uF pk , where p is prime, k ∈ N and u 2 = 0. In particular, we prove that the Gray image of a linear (1 − u)-cyclic code over R of length n is a distance-invariant quasicyclic code of index p k−1 and length p k n over F pk . We also prove that if (n, p) = 1, then every code of length p k n over F pk which is the Gray image of a linear cyclic code of length n over R is permutation-equivalent to a quasicyclic code of index p k−1 . c 2008 Elsevier Ltd. All rights reserved. Keywords: Cyclic and quasicyclic codes; Gray map; Finite rings 1. Preliminaries Let p be prime and k ∈ N. Let R be the ring F pk +uF pk , where u 2 = 0 and F pk = G F( p k ). Then R is a finite chain ring with maximal ideal u R and residue field F pk . Let C be a code of length n over R, and P(C) be its polynomial representation, i.e., ) ( n−1 X i ri x | (r0 , . . . , rn−1 ) ∈ C . P(C) = i=0 Let σ and ν be maps from R n to R n given by σ (r0 , r1 , . . . , rn−1 ) = (rn−1 , r0 , . . . , rn−2 ) and ν(r0 , r1 , . . . , rn−1 ) = ((1 − u)rn−1 , r0 , . . . , rn−2 ). Then C is said to be cyclic if σ (C) = C, and (1 − u)-cyclic if ν(C) = C. It is known that: Theorem 1.1. A code C of length n over R is cyclic if and only if P(C) is an ideal of R[x]/hx n − 1i. ∗ Corresponding address: School of Mathematics and Statistics, The University of Western Australia, Perth, Australia. E-mail addresses: mcamarra@maths.uwa.edu.au, carmen amarra@yahoo.com (M.C.V. Amarra), fidel@math.upd.edu.ph (F.R. Nemenzo). 0893-9659/$ - see front matter c 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2007.07.035 1130 M.C.V. Amarra, F.R. Nemenzo / Applied Mathematics Letters 21 (2008) 1129–1133 Theorem 1.2. A code C of length n over R is (1 − u)-cyclic if and only if P(C) is an ideal of R[x]/hx n − (1 − u)i. 
 k−1 pn pk n Let a ∈ F pk , with a = (a0 , . . . , a pk n−1 ) = a (0) | . . . | a p −1 , a (i) ∈ F pk for all i = 0, . . . , p k−1 − 1. Let σ ⊗p pk n k−1 pk n be the map from F pk to F pk given by 
 k−1 k−1 σ ⊗ p (a) = σ̃ (a (0) ) | . . . | σ̃ (a p −1 ) , pn where σ̃ is the usual cyclic shift (c0 , c1 , . . . c pn−1 ) 7→ (c pn−1 , c0 , . . . , c pn−2 ) on F pk . A code C˜ of length p k n over k−1 ˜ = C. ˜ F pk is said to be quasicyclic of index p k−1 if σ ⊗ p (C) In [1], a homogeneous weight on arbitrary finite chain rings is defined; we give it here for the case of the ring F pk + uF pk . The homogeneous weight whom (r ) of r ∈ R is given by  k  p − 1 if r ∈ R \ Ru whom (r ) = p k if r ∈ Ru \ {0}  0 otherwise. This extends to a weight function in R n : if r = (r0 , . . . , rn−1 ) ∈ R n , then whom (r) := n−1 X whom (ri ). i=0 The homogeneous distance dhom (x, y) between any distinct vectors x, y ∈ R n is defined to be whom (x − y). 2. Gray images of (1 − u)-cyclic codes over F pk + uF pk Any element ǫ ∈ Z pk has p-adic representation γ0,ǫ + γ1,ǫ p + · · · + γ(k−1),ǫ p k−1 , where γi,ǫ ∈ {0, 1, 2, . . . , p − 1}. If α is a fixed primitive element of F pk , then corresponding to every ǫ ∈ Z pk is an element αǫ ∈ F pk , given by αǫ := γ0,ǫ + γ1,ǫ α + · · · + γ(k−1),ǫ α k−1 . The Gray map Φ on R, which is a special case of the Gray map defined in [1], is given by pk n Φ : R n −→ F pk x + yu 7−→ (y, α1 x ⊕ y, . . . , α pk −1 x ⊕ y), pk n where ⊕ is componentwise addition in F pk . The Gray map Φ is an isometry from (R n , dhom ) to F pk under the Hamming distance. It is also clear that Φ preserves linearity of codes. From the above definitions, we have: Proposition 2.1. Φ ◦ ν = σ ⊗p k−1 ◦ Φ. Proof. Let r = (r0 , . . . , rn−1 ) ∈ R n and x = (x0 , x1 , . . . , xn−1 ), y = (y0 , y1 , . . . , yn−1 ) ∈ Fnpk such that r = x + yu. k−1 Let Φ(r) = (a0 , . . . , a pk n−1 ). Then σ ⊗ p (Φ(r)) = (b0 , . . . , b pk n−1 ), where  a(ǫ+ p−1)n+(n−1) if j = 0, γ0,ǫ = 0 if j = 0, γ0,ǫ 6= 0 bǫn+ j = a(ǫ−1)n+(n−1)  aǫn+ j−1 if j 6= 0. M.C.V. Amarra, F.R. Nemenzo / Applied Mathematics Letters 21 (2008) 1129–1133 1131 On the other hand, ν(r) = ((1 − u)rn−1 , r0 , . . . rn−2 ) , where (1 − u)rn−1 = xn−1 + u(−xn−1 ⊕ yn−1 ). Then Φ (ν(r)) = (c0 , . . . , c pk n−1 ), where  ! k−1 X   i  γi,ǫ α − 1 xn−1 ⊕ yn−1 , if j = 0   i=0 ! cǫn+ j = k−1 X   i  γi,ǫ α x j−1 ⊕ y j−1 , if j 6= 0   i=0 !  X k−1   γi,ǫ α i + p − 1 xn−1 ⊕ yn−1 , if j = 0, γ0,ǫ = 0     i=0  !  k−1  X i = γi,ǫ α − 1 xn−1 ⊕ yn−1 , if j = 0, γ0,ǫ 6= 0   i=0  !   k−1 X   i   γi,ǫ α x j−1 ⊕ y j−1 , if j 6= 0. i=0 The conclusion follows.  As a consequence: Theorem 2.2. A code C of length n over R is (1 − u)-cyclic if and only if Φ(C) is quasicyclic of index p k−1 and length p k n over F pk . Proof. Suppose C is (1 − u)-cyclic. Then σ ⊗p k−1 (Φ(C)) = Φ(ν(C)) = Φ(C), so Φ(C) is quasicyclic of index p k−1 . Conversely, if Φ(C) is quasicyclic of index p k−1 , then Φ(ν(C)) = σ ⊗ p k−1 (Φ(C)) = Φ(C). Since Φ is injective, it follows that ν(C) = C.  3. Gray images of cyclic codes over F pk + uF pk Suppose that (n, p) = 1. Following [2], let us have n ′ ∈ {0, 1, . . . , p − 1} such that nn ′ ≡ 1(mod p) and β = 1 + n ′ u. If µ is the map
µ : R[x]/(x n − 1) −→ R[x]/ x n − (1 − u) r (x) 7−→ r (βx) then µ is a ring isomorphism. Hence I is an ideal of R[x]/hx n −1i if and only if µ(I ) is an ideal of R[x]/hx n −(1−u)i. If µ is the map µ : R n −→ R n   r 7−→ r0 , βr1 , β 2r2 , . . . , β n−1rn−1 then it also follows that: Proposition 3.1. The set C ⊆ R n is a linear cyclic code if and only if µ(C) is a linear (1 − u)-cyclic code. We now define the permutation π ⊗ p follows: k−1 , which is an extension of the Nechaev permutation introduced in [4], as 1132 M.C.V. Amarra, F.R. Nemenzo / Applied Mathematics Letters 21 (2008) 1129–1133  0 For c = c( p ) | . . . |c π⊗p k−1 where p k−1
 pk n ∈ F pk ,    0 (c) := π c( p ) | . . . | π c π(a) = aτ (0) , . . . , aτ ( pn−1) pn p k−1
 ,
for a = (a0 , . . . , a pn−1 ) ∈ F pk , where τ (γ n + j) = (γ + jn ′ ) p n + j, 0 ≤ γ ≤ p − 1, 0 ≤ j ≤ n − 1, and (γ + jn ′ ) p is the least residue of γ + jn ′ modulo p. Proposition 3.2. Φ ◦ µ = π⊗p k−1 ◦ Φ. Proof. Let r = (r0 , . . . , rn−1 ) ∈ R n . If Φ(r) = (a0 , . . . , a pk n−1 ), then π ⊗ p k−1 (Φ(r)) = (b0 , . . . , b pk n−1 ), where bǫn+ j = a[γ (ǫ) +(γ0,ǫ + jn ′ ) p ]n+ j # " k−1 X i ′ γi,ǫ α + (γ0,ǫ + jn ) p x j ⊕ y j . = i=1
On the other hand, µ(r) = r0 , βr1 , . . . , β n−1rn−1 , where β j r j = (1 + n ′ u) j r j = x j + u( jn ′ x j ⊕ y j ). Hence Φ(µ(r)) = (c0 , c1 , . . . , c pk n−1 ), where ! k−1 X i γi,ǫ α x j ⊕ ( jn ′ x j ⊕ y j ) cǫn+ j = i=0 = " k−1 X # γi,ǫ α i + (γ0,ǫ + jn ′ ) p x j ⊕ y j . i=1 The conclusion follows.  Corollary 3.3. If C̃ is the Gray image of a linear cyclic code of length n over R, then C̃ is equivalent to a quasicyclic code of index p k−1 and length p k n over F pk . Proof. From Proposition 3.1, a code C of length n over R is linear cyclic if and only if µ(C) is linear (1 − u)-cyclic. From Theorem 2.2, this is so if and only if Φ(µ(C)) is a linear quasicyclic code of index p k−1 over F pk , that is, if and only if π ⊗ p k−1 (Φ(C)) is linear quasicyclic of index p k−1 over F pk .  The following example illustrates the above results. Computations were done using MAGMA. Example. Let F4 = {0, 1, ω, 1 + ω = ω2 }. Consider x 3 − 1. In F4 + uF4 , x 3 − 1 = (x + 1)(x + ω)(x + ω2 ). Applying the map µ : x 7→ (1 + u)x, we obtain x 3 − (1 − u) = [x + (u + 1)][x + (ωu + ω)][x + (ω2 u + ω2 )] = f 1 (x) f 2 (x) f 3 (x). M.C.V. Amarra, F.R. Nemenzo / Applied Mathematics Letters 21 (2008) 1129–1133 1133 1. Let C1 be the (1 − u)-cyclic code generated by a, where a(x) = f 1 (x) f 2 (x) = x 2 + (ω2 u + ω2 )x + 1. The code Φ(C1 ) is a [12, 2, 9] quaternary code, which is an optimal code. 2. Let a(x) be as in the above, b(x) = u f 1 (x) = ux + u, and C2 be the (1 − u)-cyclic code over F4 + uF4 generated by a and b. The code Φ(C2 ) is a [12, 3, 8] quaternary code, which is an optimal code. References [1] M. Greferath, S.E. Schmidt, Gray isometries for finite chain rings and a nonlinear ternary (36, 312 , 15) code, IEEE Trans. Inform. Theory 45 (1999) 2522–2524. [2] S. Ling, J. Blackford, Z pk+1 -linear codes, IEEE Trans. Inform. Theory 48 (2002) 2592–2605. [3] J.F. Qian, L.N. Zhang, S.X. Zhu, (1 + u)-constacyclic and cyclic codes over F2 + uF2 , Appl. Math. Lett. 19 (2006) 820–823. [4] J. Wolfmann, Negacyclic and cyclic codes over Z4 , IEEE Trans. Inform. Theory 45 (1999) 2527–2532.