arXiv subjects

Hai Q. Dinh

Publications and source records attributed to Hai Q. Dinh.

7 recordsLinked to original sources

arXiv2019

An efficient method to construct self-dual cyclic codes of length $p^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$

Let $p$ be an odd prime number, $\mathbb{F}_{p^m}$ be a finite field of cardinality $p^m$ and $s$ a positive integer. Using some combinatorial identities, we obtain certain properties for Kronecker product of matrices over $\mathbb{F}_p$ with a specific type. On that basis, we give an explicit representation and enumeration for all distinct self-dual cyclic codes of length $p^s$ over the finite chain ring $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$ $(u^2=0)$. Moreover, We provide an efficient method to construct every self-dual cyclic code of length $p^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$ precisely.

cs.IT↗

arXiv2019

Construction and enumeration for self-dual cyclic codes of even length over $\mathbb{F}_{2^m} + u\mathbb{F}_{2^m}$

Let $\mathbb{F}_{2^m}$ be a finite field of cardinality $2^m$, $R=\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ $(u^2=0)$ and $s,n$ be positive integers such that $n$ is odd. In this paper, we give an explicit representation for every self-dual cyclic code over the finite chain ring $R$ of length $2^sn$ and provide a calculation method to obtain all distinct codes. Moreover, we obtain a clear formula to count the number of all these self-dual cyclic codes. As an application, self-dual and $2$-quasi-cyclic codes over $\mathbb{F}_{2^m}$ of length $2^{s+1}n$ can be obtained from self-dual cyclic code over $R$ of length $2^sn$ and by a Gray map preserving orthogonality and distances from $R$ onto $\mathbb{F}_{2^m}^2$.

cs.IT↗

arXiv2019

Explicit representation for a class of Type 2 constacyclic codes over the ring $\mathbb{F}_{2^m}[u]/\langle u^{2λ}\rangle$ with even length

Let $\mathbb{F}_{2^m}$ be a finite field of cardinality $2^m$, $λ$ and $k$ be integers satisfying $λ,k\geq 2$ and denote $R=\mathbb{F}_{2^m}[u]/\langle u^{2λ}\rangle$. Let $δ,α\in \mathbb{F}_{2^m}^{\times}$. For any odd positive integer $n$, we give an explicit representation and enumeration for all distinct $(δ+αu^2)$-constacyclic codes over $R$ of length $2^kn$, and provide a clear formula to count the number of all these codes. As a corollary, we conclude that every $(δ+αu^2)$-constacyclic code over $R$ of length $2^kn$ is an ideal generated by at most $2$ polynomials in the residue class ring $R[x]/\langle x^{2^kn}-(δ+αu^2)\rangle$.

cs.IT↗

arXiv2018

An explicit representation and enumeration for self-dual cyclic codes over $\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ of length $2^s$

Let $\mathbb{F}_{2^m}$ be a finite field of cardinality $2^m$ and $s$ a positive integer. Using properties for Kronecker product of matrices and calculation for linear equations over $\mathbb{F}_{2^m}$, an efficient method for the construction of all distinct self-dual cyclic codes with length $2^s$ over the finite chain ring $\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ $(u^2=0)$ is provided. On that basis, an explicit representation for every self-dual cyclic code of length $2^s$ over $\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ and an exact formula to count the number of all these self-dual cyclic codes are given.

cs.IT↗

arXiv2018

A class of repeated-root constacyclic codes over $\mathbb{F}_{p^m}[u]/\langle u^e\rangle$ of Type $2$

Let $\mathbb{F}_{p^m}$ be a finite field of cardinality $p^m$ where $p$ is an odd prime, $n$ be a positive integer satisfying ${\rm gcd}(n,p)=1$, and denote $R=\mathbb{F}_{p^m}[u]/\langle u^e\rangle$ where $e\geq 4$ be an even integer. Let $δ,α\in \mathbb{F}_{p^m}^{\times}$. Then the class of $(δ+αu^2)$-constacyclic codes over $R$ is a significant subclass of constacyclic codes over $R$ of Type 2. For any integer $k\geq 1$, an explicit representation and a complete description for all distinct $(δ+αu^2)$-constacyclic codes over $R$ of length $np^k$ and their dual codes are given. Moreover, formulas for the number of codewords in each code and the number of all such codes are provided respectively. In particular, all distinct $(δ+αu^2)$-contacyclic codes over $\mathbb{F}_{p^m}[u]/\langle u^{e}\rangle$ of length $p^k$ and their dual codes are presented precisely.

cs.IT↗

arXiv2014

Polyadic Constacyclic Codes

For any given positive integer $m$, a necessary and sufficient condition for the existence of Type I $m$-adic constacyclic codes is given. Further, for any given integer $s$, a necessary and sufficient condition for $s$ to be a multiplier of a Type I polyadic constacyclic code is given. As an application, some optimal codes from Type I polyadic constacyclic codes, including generalized Reed-Solomon codes and alternant MDS codes, are constructed.

cs.IT↗

arXiv2014

Repeated-root constacyclic codes of length $2\ell^mp^n$

For any different odd primes $\ell$ and $p$, structure of constacyclic codes of length $2\ell^mp^n$ over a finite field $\mathbb F_q$ of characteritic $p$ and their duals is established in term of their generator polynomials. Among other results, all linear complimentary dual and self-dual constacyclic codes of length $2\ell^mp^n$ over $\mathbb F_q$ are obtained.

cs.IT↗