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Register a new user$sigma$-self-orthogonal constacyclic codes of length $p^s$ over $mathbb F_{p^m}+umathbb F_{p^m}$
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Added by
Jingge Liu
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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In this paper, we study the $sigma$-self-orthogonality of constacyclic codes of length $p^s$ over the finite commutative chain ring $mathbb F_{p^m} + u mathbb F_{p^m}$, where $u^2=0$ and $sigma$ is a ring automorphism of $mathbb F_{p^m} + u mathbb F_{p^m}$. First, we obtain the structure of $sigma$-dual code of a $lambda$-constacyclic code of length $p^s$ over $mathbb F_{p^m} + u mathbb F_{p^m}$. Then, the necessary and sufficient conditions for a $lambda$-constacyclic code to be $sigma$-self-orthogonal are provided. In particular, we determine the $sigma$-self-dual constacyclic codes of length $p^s$ over $mathbb F_{p^m} + u mathbb F_{p^m}$. Finally, we extend the results to constacyclic codes of length $2 p^s$.
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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.
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 $mathbb{F}_q+umathbb{F}_q$.
BCH codes are an interesting class of cyclic codes due to their efficient encoding and decoding algorithms. In many cases, BCH codes are the best linear codes. However, the dimension and minimum distance of BCH codes have been seldom solved. Until now, there have been few results on BCH codes over $gf(q)$ with length $q^m+1$, especially when $q$ is a prime power and $m$ is even. The objective of this paper is to study BCH codes of this type over finite fields and analyse their parameters. The BCH codes presented in this paper have good parameters in general, and contain many optimal linear codes.
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.
We study the mod-p cohomology of the group Out(F_n) of outer automorphisms of the free group F_n in the case n=2(p-1) which is the smallest n for which the p-rank of this group is 2. For p=3 we give a complete computation, at least above the virtual cohomological dimension of Out(F_4) (which is 5). More precisley, we calculate the equivariant cohomology of the p-singular part of outer space for p=3. For a general prime p>3 we give a recursive description in terms of the mod-p cohomology of Aut(F_k) for k less or equal to p-1. In this case we use the Out(F_{2(p-1)})-equivariant cohomology of the poset of elementary abelian p-subgroups of Out(F_n).
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