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In this paper, we construct a class of Hermitian self-dual 2-quasi-abelian codes over a finite field. Based on counting the number of such codes and estimating the number of the codes in this class whose relative minimum weights are small, we prove that the class of Hermitian self-dual 2-quasi-abelian codes over any finite field is asymptotically good. The existence of such codes is unconditional, which is different from the case of Euclidean self-dual 2-quasi-abelian codes over a special finite field. [ABSTRACT FROM AUTHOR]

Galois hulls of MDS codes can be applied to construst MDS entanglement-assisted quantum error-correcting codes (EAQECCs). Goppa codes and expurgated Goppa codes (resp., extended Goppa codes) over F q m are GRS codes (resp., extended GRS codes) when m = 1. In this paper, we investigate the Galois dual codes of a special kind of Goppa codes and related codes and provide a necessary and sufficient condition for the Galois dual codes of such codes to be Goppa codes and related codes. Then we determine the Galois hulls of the above codes. In particular, we completely characterize Galois LCD, Galois self-orthogonal, Galois dual-containing and Galois self-dual codes among such family of codes. Moreover, we apply the above results to EAQECCs. [ABSTRACT FROM AUTHOR]

Let F q be the finite field with q elements and C n be the cyclic group of order n , where n is a positive integer relatively prime to q . Let H , K be subgroups of C n such that H is a proper subgroup of K . In this note, the weight distributions of the cyclic codes of length n over F q with generating idempotents K ˆ and e H , K = H ˆ − K ˆ are explicitly determined, where K ˆ = 1 / | K | ∑ g ∈ K g and H ˆ = 1 / | H | ∑ g ∈ H g . Our result naturally gives a new characterization of a theorem by Sharma and Bakshi [18] that determines the weight distribution of all irreducible cyclic codes of length p m over F q , where p is an odd prime and q is a primitive root modulo p m . Finally, two examples are presented to illustrate our results. [ABSTRACT FROM AUTHOR]

Abstract: An equivalence relation called isometry is introduced to classify constacyclic codes over a finite field; the polynomial generators of constacyclic codes of length are characterized, where p is the characteristic of the finite field and ℓ is a prime different from p. [Copyright &y& Elsevier]