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In this paper, we propose a family of quantum synchronizable codes from repeated-root cyclic codes and constacyclic codes. This family of quantum synchronizable codes are based on (λ(u + v)|u - v) construction which is constructed from constacyclic codes. Under this construction, we enrich the varieties of valid quantum synchronizable codes. We also prove that the obtained quantum synchronizable codes can achieve maximum synchronization error tolerance. Furthermore, quantum synchronizable codes based on (λ(u + v)|u - v) construction are shown to be able to have a better capability in correcting bit errors than those from projective geometry codes.

Symbol-pair codes can be used to guard against pair errors over symbol-pair read channels. One of the central themes in symbol-error correction is the construction of maximal distance separable (MDS) symbol-pair codes that possess the largest possible pair-error correcting performance. In this letter, we construct three new MDS symbol-pair codes with minimum pair-distance six or seven through repeated-root constacyclic codes. In contrast with classical MDS codes, they have relatively large length. [ABSTRACT FROM PUBLISHER]

In this paper, we explicitly determine the generator polynomials of all repeated-root constacyclic codes of length n l p s over F q and their dual codes, where l is an odd prime different from p , and n is an odd prime different from both l and p such that n = 2 h + 1 for some prime h or h = 1 . Moreover, we give all self-dual cyclic codes of length n l p s over F q and their enumeration. All linear complementary dual (LCD) cyclic codes of length n l p s over F q and their enumeration are also obtained. [ABSTRACT FROM AUTHOR]

Let p ≠ 3 be any prime and l ≠ 3 be any odd prime with gcd ⁡ ( p , l ) = 1 . The multiplicative group F q ⁎ = 〈 ξ 〉 can be decomposed into mutually disjoint union of gcd ⁡ ( q − 1 , 3 l p s ) cosets over the subgroup 〈 ξ 3 l p s 〉 , where ξ is a primitive ( q − 1 ) th root of unity. We classify all repeated-root constacyclic codes of length 3 l p s over the finite field F q into some equivalence classes by this decomposition, where q = p m , s and m are positive integers. According to these equivalence classes, we explicitly determine the generator polynomials of all repeated-root constacyclic codes of length 3 l p s over F q and their dual codes. Self-dual cyclic codes of length 3 l p s over F q exist only when p = 2 . We give all self-dual cyclic codes of length 3 ⋅ 2 s l over F 2 m and their enumeration. We also determine the minimum Hamming distance of these codes when gcd ⁡ ( 3 , q − 1 ) = 1 and l = 1 . [ABSTRACT FROM AUTHOR]

Abstract: Let be a finite field with elements, where p is an odd prime and . In this paper, we explicitly determine all the μ-constacyclic codes of length over , when the order of μ is a power of 2. We further obtain all the self-dual negacyclic codes of length over and give some illustrative examples. All the repeated-root λ-constacyclic codes of length over are also determined for any nonzero λ in . As examples all the 2-constacyclic, 3-constacyclic codes of length over and all the 3-constacyclic, 5-constacyclic codes of length over for , are derived. [Copyright &y& Elsevier]

Let R be a finite commutative chain ring and γ be a fixed generator of the maximal ideal of R. For any unit w in R , (1 + w γ) -constacyclic codes over R as a generalization of negacyclic codes over Z 4 are a class of important linear codes. In this paper, based on algebraic structure, the generator polynomials of all torsion codes of (1 + w γ) -constacyclic codes of any length over R are first given. Then, by using these torsion codes, we completely determine the depth spectrums of (1 + w γ) -constacyclic codes over R of any length. [ABSTRACT FROM AUTHOR]