Constacyclic codes over finite commutative semi-simple rings

Finite commutative semi-simple rings are direct sum of finite fields. In this study, we investigate the algebraic structure of λ-constacyclic codes over such finite semi-simple rings. Among others, necessary and sufficient conditions for the existence of self-dual, LCD, and Hermitian dual-containing λ-constacyclic codes over finite semi-simple rings are provided. Using the CSS and Hermitian constructions, quantum MDS codes over finite semi-simple rings are constructed.     

On the symbol-pair distances of repeated-root constacyclic codes of length 2ps

Let $p$ be an odd prime, and $\lambda$ be a nonzero element of the finite field ${\mathbb{F}}_{p^m}$. The $\lambda$-constacyclic codes of length $2p^s$ over ${\mathbb{F}}_{p^m}$ are classified as the ideals of quotient ring $\frac{{\mathbb{F}}_{p^m}\left[x\right]}{〈x^{2p^s}?\lambda 〉}$ in terms of their generator polynomials. Based on these generator polynomials, the symbol-pair distances of all such $\lambda$-constacyclic codes of length $2p^s$ are obtained in this paper. As an application, all MDS symbol-pair constacyclic codes of length $2p^s$ over ${\mathbb{F}}_{p^m}$ are established, which produce many new MDS symbol-pair codes with good parameters.     

Duals of constacyclic codes over finite local Frobenius non-chain rings of length 4

Duals of constacyclic codes over a finite local Frobenius non-chain ring of length 4, the length of which is relatively prime to the characteristic of the residue field of the ring are determined. Generators for the dual code are obtained from those of the original constacyclic code. In some cases self-dual codes are determined.     

On the Hamming distances of repeated-root constacyclic codes of length 4ps

Let $p$ be an odd prime, $s$, $m$ be positive integers, $\gamma ,\lambda$ be nonzero elements of the finite field ${\mathbb{F}}_{p^m}$ such that ${\gamma }^{p^s}=\lambda$. In this paper, we show that, for any positive integer $\eta$, the Hamming distances of all repeated-root $\lambda$-constacyclic codes of length $\eta p^s$ can be determined by those of certain simple-root $\gamma$-constacyclic codes of length $\eta$. Using this result, Hamming distances of all constacyclic codes of length $4p^s$ are obtained. As an application, we identify all MDS $\lambda$-constacyclic codes of length $4p^s$.     

Constacyclic codes over finite local Frobenius non-chain rings with nilpotency index 3

The main results of this paper are in two directions. First, the family of finite local Frobenius non-chain rings of length 4 (hence of nilpotency index 3) is determined. As a by-product all finite local Frobenius non-chain rings with $p^4$ elements, (p a prime) are given. Second, the number and structure of γ-constacyclic codes over finite local Frobenius non-chain rings with nilpotency index 3, of length relatively prime to the characteristic of the residue field of the ring, are determined.