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Let $n (>3)$ be a prime number and ${\mathbb {F}}_{2^{n}}$ a finite field of $2^{n}$ elements. Let $L ={\mathbb {F}}_{2^{n}}\cup \{\infty \}$ be the support set and $g(x)$ an irreducible polynomial of degree 6 over ${\mathbb {F}}_{2^{n}}$. In this paper, we obtain an upper bound on the number of extended irreducible binary Goppa codes $\Gamma (L, g)$ of degree 6 and length $2^{n}+1$. [ABSTRACT FROM AUTHOR]

In this paper, we always assume that p and n > 3 are two distinct odd primes, m ≥ 2 is a positive integer, and gcd ⁡ (p m , 2 n (2 2 n − 1)) ≠ 1. Set r = p 2 or r = p 3 , we determine upper bounds on the number of inequivalent extended binary irreducible Goppa codes Γ (L , g) of degree r and length 2 n + 1 , where L = F 2 n ∪ { ∞ } is the support set and each g (x) is an irreducible polynomial of degree r over F 2 n . [ABSTRACT FROM AUTHOR]

The codes in the title are studied with respect to existence, enumeration and asymptotic performance. Their Gray images are shown to satisfy a modified Gilbert-Varshamov bound. Explicit counting formulas are derived. Examples of modest lengths are given where Gray images of LCD Z 4 -codes outperform the best known binary linear LCD codes. [ABSTRACT FROM AUTHOR]

We study double circulant LCD codes over ℤ p 2 for all odd primes p , and self-dual double circulant codes over ℤ p 2 for primes p ≡ 1 (mod   4). We derive exact enumeration formulae, and asymptotic lower bounds on the minimum distance of the p -ary images of these codes by the classical Gray maps. [ABSTRACT FROM AUTHOR]

Constant dimension codes (CDCs), as special subspace codes, have attracted extensive attention due to their application at random network coding. In this paper, we present some variations on the insert construction method by changing the positions of some matrices. Those variations help us to construct more types of CDCs. By restricting the rank of matrices in rank metric codes, We can get that the union of these codes is still a CDC and the minimum subspace distance can be preserved. We generalize the previous results in He et al. (IEEE Commun Lett 25(5):1422–1426, 2021) and Lao et al. (arXiv:2008.09944v1, 2020) and obtain some new lower bounds of CDCs. [ABSTRACT FROM AUTHOR]

Let q be a prime power and m a positive integer. Suppose that a ≥ 2 is a factor of q m − 1 such that m is the multiplicative order of q modulo n : = q m − 1 a. Firstly, for m odd and m even, we present some necessary and sufficient conditions on dual-containing non-primitive BCH codes of length n with the maximum designed distances over F q , respectively. The results show that the maximum designed distances of dual-containing BCH codes in this paper are larger than those in [2, Theorem 3]. Secondly, we explicitly determine the dimensions of some dual-containing non-primitive BCH codes of length n , and finally, we construct some new quantum codes with relatively large minimum distances. [ABSTRACT FROM AUTHOR]

Twisted generalized Reed-Solomon (TGRS) codes are a family of codes that contains a large number of maximum distance separable (MDS) codes that are non-equivalent to generalized Reed-Solomon (GRS) codes. In this paper, we characterize a sufficient and necessary condition that a twisted Reed-Solomon (TRS) code with two twists is MDS; give a sufficient and necessary condition that a TGRS code with two twists is self-dual, and present some constructions of self-dual TGRS codes. These self-dual codes are MDS, NMDS or 2-MDS. Furthermore, we study the non-GRS properties of TGRS codes with two twists and prove that these codes are non-GRS in most cases. [ABSTRACT FROM AUTHOR]