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Register a new userSelf-Dual Skew Cyclic Codes over $mathbb{F}_{q}+umathbb{F}_{q}$
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Added by
Kenza Guenda
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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In this paper, we give conditions for the existence of Hermitian self-dual $Theta-$cyclic and $Theta-$negacyclic codes over the finite chain ring $mathbb{F}_q+umathbb{F}_q$. By defining a Gray map from $R=mathbb{F}_q+umathbb{F}_q$ to $mathbb{F}_{q}^{2}$, we prove that the Gray images of skew cyclic codes of odd length $n$ over $R$ with even characteristic are equivalent to skew quasi-twisted codes of length $2n$ over $mathbb{F}_q$ of index $2$. We also extend an algorithm of Boucher and Ulmer cite{BF3} to construct self-dual skew cyclic codes based on the least common left multiples of non-commutative polynomials over $mathbb{F}_q+umathbb{F}_q$.
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In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring $R=F_{q}+vF_{q}+v^{2}F_{q}$, where $v^{3}=v$, for $q$ odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over $R$. Further, we give bounds on the minimum distance of LCD codes over $F_q$ and extend these to codes over $R$.
We apply quantum Construction X on quasi-cyclic codes with large Hermitian hulls over $mathbb{F}_4$ and $mathbb{F}_9$ to derive good qubit and qutrit stabilizer codes, respectively. In several occasions we obtain quantum codes with stricly improved parameters than the current record. In numerous other occasions we obtain quantum codes with best-known performance. For the qutrit ones we supply a systematic construction to fill some gaps in the literature.
Let $mathbb{F}_p$ be a finite field and $u$ be an indeterminate. This article studies $(1-2u^k)$-constacyclic codes over the ring $mathcal{R}=mathbb{F}_p+umathbb{F}_p+u^2mathbb{F}_p+u^{3}mathbb{F}_{p}+cdots+u^{k}mathbb{F}_{p}$ where $u^{k+1}=u$. We illustrate the generator polynomials and investigate the structural properties of these codes via decomposition theorem.
In this paper, the investigation on the algebraic structure of the ring $frac{mathbb{F}_q[v]}{langle,v^q-v,rangle}$ and the description of its automorphism group, enable to study the algebraic structure of codes and their dual over this ring. We explore the algebraic structure of skew-constacyclic codes, by using a linear Gray map and we determine their generator polynomials. Necessary and sufficient conditions for the existence of self-dual skew cyclic and self-dual skew negacyclic codes over $frac{mathbb{F}_q[v]}{langle,v^q-v,rangle}$ are given.
We introduce a consistent and efficient method to construct self-dual codes over $GF(q)$ with symmetric generator matrices from a self-dual code over $GF(q)$ of smaller length where $q equiv 1 pmod 4$. Using this method, we improve the best-known minimum weights of self-dual codes, which have not significantly improved for almost two decades. We focus on a class of self-dual codes, including double circulant codes. Using our method, called a `symmetric building-up construction, we obtain many new self-dual codes over $GF(13)$ and $GF(17)$ and improve the bounds of best-known minimum weights of self-dual codes of lengths up to 40. Besides, we compute the minimum weights of quadratic residue codes that were not known before. These are: a [20,10,10] QR self-dual code over $GF(23)$, two [24,12,12] QR self-dual codes over $GF(29)$ and $GF(41)$, and a [32,12,14] QR self-dual codes over $GF(19)$. They have the highest minimum weights so far.
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