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Register a new user$(1-2u^k)$-constacyclic codes over $mathbb{F}_p+umathbb{F}_p+u^2mathbb{F}_+u^{3}mathbb{F}_{p}+dots+u^{k}mathbb{F}_{p}$
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Added by
Zahid Raza
Publication date
2015
fields
Informatics Engineering
and research's language is
English
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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.
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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 $mathbb{F}_q+umathbb{F}_q$.
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.
Determinants with Legendre symbol entries have close relations with character sums and elliptic curves over finite fields. In recent years, Sun, Krachun and his cooperators studied this topic. In this paper, we confirm some conjectures posed by Sun and investigate some related topics. For instance, given any integers $c,d$ with $d e0$ and $c^2-4d e0$, we show that there are infinitely many odd primes $p$ such that $$detbigg[left(frac{i^2+cij+dj^2}{p}right)bigg]_{0le i,jle p-1}=0,$$ where $(frac{cdot}{p})$ is the Legendre symbol. This confirms a conjecture of Sun.
Let $mathbb{F}_{2^m}$ be a finite field of $2^m$ elements, and $R=mathbb{F}_{2^m}[u]/langle u^krangle=mathbb{F}_{2^m}+umathbb{F}_{2^m}+ldots+u^{k-1}mathbb{F}_{2^m}$ ($u^k=0$) where $k$ is an integer satisfying $kgeq 2$. For any odd positive integer $n$, an explicit representation for every self-dual cyclic code over $R$ of length $2n$ and a mass formula to count the number of these codes are given first. Then a generator matrix is provided for the self-dual and $2$-quasi-cyclic code of length $4n$ over $mathbb{F}_{2^m}$ derived by every self-dual cyclic code of length $2n$ over $mathbb{F}_{2^m}+umathbb{F}_{2^m}$ and a Gray map from $mathbb{F}_{2^m}+umathbb{F}_{2^m}$ onto $mathbb{F}_{2^m}^2$. Finally, the hull of each cyclic code with length $2n$ over $mathbb{F}_{2^m}+umathbb{F}_{2^m}$ is determined and all distinct self-orthogonal cyclic codes of length $2n$ over $mathbb{F}_{2^m}+umathbb{F}_{2^m}$ are listed.
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