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A class of self-dual quasi-abelian codes of index 2 over any finite field $F$ is introduced. By counting the number of such codes and the number of the codes in this class whose relative minimum weights are small, such codes are proved to be asymptotically good provided −1 is a square in $F$. Moreover, a class of self-orthogonal quasi-abelian codes of index 2 is defined; and such codes always exist. In a way similar to that for self-dual quasi-abelian codes of index 2, it is proved that these self-orthogonal quasi-abelian codes of index 2 are asymptotically good. [ABSTRACT FROM AUTHOR]

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]

Bazzi and Mitter showed that binary dihedral group codes are asymptotically good. In this paper we prove that the dihedral group codes over any finite field with strong duality property are asymptotically good. If the characteristic of the field is even, self-dual dihedral group codes are asymptotically good. If the characteristic of the field is odd, maximal self-orthogonal dihedral group codes and LCD dihedral group codes are asymptotically good. [ABSTRACT FROM AUTHOR]

The Born approximation and the Born sequence are considered for hyperbolic equations when we perturb their leading parts. The Born approximation is a finite successive approximation such as the finite terms Neumann series for the solution of a hyperbolic equation in terms of the smallness of the perturbation and if the successive approximation is infinitely many times, then we have the Born series. Due to the so called regularity loss for solutions of hyperbolic equations, we need to assume that data such as the inhomogeneous term of the equation, Cauchy datum and boundary datum are C ∞ , and also they satisfy the compatibility condition of any order in order to define the Born series. Otherwise we need to smooth each term of the Born series. The convergence of the Born series and the Born series with smoothing are very natural questions to be asked. Also giving an estimate of approximating the solution for finite terms Born series is also an important question in practice. The aims of this paper are to discuss about these questions. We would like to emphasize that we found a small improvement in the usual energy estimate for solutions of an initial value problem for a hyperbolic equation, which is very useful for our aims. Since the estimate of approximation is only giving the worst estimate for the approximation, we also provide some numerical studies on these questions which are very suggestive for further theoretical studies on the Born approximation for hyperbolic equations. [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]

Symbol-pair codes introduced by Cassuto and Blaum (2010) are designed to protect against pair errors in symbol-pair read channels. The higher the minimum pair distance, the more pair errors the code can correct. Maximum distance separable (MDS) symbol-pair codes are optimal in the sense that pair distance cannot be improved for given length and code size. The contribution of this paper is twofold. First, we present three lower bounds for the minimum pair distance of constacyclic codes, the first two of which generalize the previously known results due to Cassuto and Blaum (2011) and Kai et al. (2015). The third one exhibits a lower bound for the minimum pair distance of repeated-root cyclic codes. Second, we obtain new MDS symbol-pair codes with minimum pair distance seven and eight through repeated-root cyclic codes. [ABSTRACT FROM AUTHOR]

External difference families (EDFs) are a type of combinatorial designs that originated from cryptography. Many combinatorial objects are closely related to EDFs, such as difference sets, difference families, almost difference sets, and difference systems of sets. Constructing EDFs is thus of significance in theory and practice. In this paper, earlier ideas of constructing EDFs proposed by Chang and Ding (2006), and Huang and Wu (2009), are further explored. Consequently, new infinite classes of EDFs are obtained and some previously known results are extended. [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]

For a $q $ -ary random quasi-Abelian code with fixed coindex and constant rate r $ , it is shown that the Gilbert–Varshamov (GV)-bound is a threshold point: if r $ is less than the GV-bound at , then the probability of the relative distance of the random code being greater than $\delta $ approaches 1 as the index goes to infinity; whereas, if $r $ is bigger than the GV-bound at $\delta $ , then the probability approaches 0. As a corollary, there exist numerous asymptotically good quasi-Abelian codes attaining the GV-bound. [ABSTRACT FROM PUBLISHER]