Bounds for the minimum distance in constacyclic codes

In algebraic coding theory it is common practice to require that (n, q) = 1, where n is the word length and F = GF(q) is the alphabet. In this paper, which is about constacyclic codes, we shall stick to this practice too. Since linear codes have the structure of linear subspaces of F , an alternative description of constacyclic codes in terms of linear algebra appears to be another quite natural approach. Due to this description we derive lower bounds for the minimum distance of constacyclic codes that are generalizations of the well known BCH bound, the Hartmann-Tzeng bound and the Roos bound. Definition 1. Let a be a nonzero element of F = GF(q). A code C of length n over F is called constacyclic with respect to a, if whenever x = (c1, c2, . . . , cn) is in C, so is y = (acn, c1, . . . , cn−1). Let a be a nonzero element of F and let ψa : { Fn → Fn (x1, x2, . . . , xn) 7→ (axn, x1, . . . , xn−1) . Then ψa ∈ HomFn and it has the following matrix Bn(a) = Bn =   0 0 0 . . . a 1 0 0 . . . 0 0 1 0 . . . 0 .. .. .. . . . .. 0 0 0 . . . 0   with respect to the standard basis e = (e1, e2, . . . , en). The characteristic polynomial of Bn is fBn(x) = (−1)n(xn − a). We shall denote it by f(x). We assume that (n, q) = 1. The polynomial f(x) has no multiple roots and splits into distinct irreducible monic factors f(x) = (−1)f1(x) . . . ft(x). Let Ui = Ker fi(ψa), i = 1, . . . , n. For the proof of the following theorem we refer to [1]. Radkova, van Zanten 237 Theorem 1. Let C be a linear constacyclic code of length n over F. Then the following facts hold. 1) C is a constacyclic code iff C is a ψa−invariant subspace of Fn; 2) C = Ui1 ⊕ · · · ⊕Uis for some minimal ψa−invariant subspaces Uir of Fn and k := dim F C = ki1 + · · ·+ kis , where kir is the dimension of Uir ; 3) fψa|C (x) = (−1)fi1(x) . . . fis(x) = g(x); 4) c ∈ C iff g(Bn)c = 0; 5) the polynomial g(x) has the smallest degree with respect to property 4); 6) r (g(Bn)) = n − k, where r (g(Bn)) = n − k is the rank of the matrix g(Bn). Let K = GF(qm) be the splitting field of the polynomial f(x) = (−1)n(xn− a) over F, where 0 6= a ∈ F. Let the eigenvalues of ψa be α1, . . . , αn, with αi = n √ aαi, i = 1, . . . , n, where α is a primitive n−th root of unity and n √a is a fixed, but otherwise arbitrary, zero of the polynomial xn − a. Let vi be the respective eigenvectors, i = 1, . . . , n. More in particular we have Bnv i = αiv t i, vi = (αi n−1, αin−2, . . . , αi, 1), i = 1, . . . , n. Let us consider the basis v = (v1, . . . ,vn) of eigenvectors of ψa. We carry out the basis transformation e → v, and obtain that D =   α1 0 . . . 0 0 α2 . . . 0 .. .. . . . .. 0 0 . . . αn   = T−1BnT,