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Register a new userMulti-twisted codes over finite fields and their dual codes
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Let $mathbb{F}_{q}$ denote the finite field of order $q,$ let $m_1,m_2,cdots,m_{ell}$ be positive integers satisfying $gcd(m_i,q)=1$ for $1 leq i leq ell,$ and let $n=m_1+m_2+cdots+m_{ell}.$ Let $Lambda=(lambda_1,lambda_2,cdots,lambda_{ell})$ be fixed, where $lambda_1,lambda_2,cdots,lambda_{ell}$ are non-zero elements of $mathbb{F}_{q}.$ In this paper, we study the algebraic structure of $Lambda$-multi-twisted codes of length $n$ over $mathbb{F}_{q}$ and their dual codes with respect to the standard inner product on $mathbb{F}_{q}^n.$ We provide necessary and sufficient conditions for the existence of a self-dual $Lambda$-multi-twisted code of length $n$ over $mathbb{F}_{q},$ and obtain enumeration formulae for all self-dual and self-orthogonal $Lambda$-multi-twisted codes of length $n$ over $mathbb{F}_{q}.$ We also derive some sufficient conditions under which a $Lambda$-multi-twisted code is LCD. We determine the parity-check polynomial of all $Lambda$-multi-twisted codes of length $n$ over $mathbb{F}_{q}$ and obtain a BCH type bound on their minimum Hamming distances. We also determine generating sets of dual codes of some $Lambda$-multi-twisted codes of length $n$ over $mathbb{F}_{q}$ from the generating sets of the codes. Besides this, we provide a trace description for all $Lambda$-multi-twisted codes of length $n$ over $mathbb{F}_{q}$ by viewing these codes as direct sums of certain concatenated codes, which leads to a method to construct these codes. We also obtain a lower bound on their minimum Hamming distances using their multilevel concatenated structure.
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In this paper, we produce new classes of MDS self-dual codes via (extended) generalized Reed-Solomon codes over finite fields of odd characteristic. Among our constructions, there are many MDS self-dual codes with new parameters which have never been reported. For odd prime power $q$ with $q$ square, the total number of lengths for MDS self-dual codes over $mathbb{F}_q$ presented in this paper is much more than those in all the previous results.
In this paper, we present three new classes of $q$-ary quantum MDS codes utilizing generalized Reed-Solomon codes satisfying Hermitian self-orthogonal property. Among our constructions, the minimum distance of some $q$-ary quantum MDS codes can be bigger than $frac{q}{2}+1$. Comparing to previous known constructions, the lengths of codes in our constructions are more flexible.
Self-dual codes over $Z_2timesZ_4$ are subgroups of $Z_2^alpha timesZ_4^beta$ that are equal to their orthogonal under an inner-product that relates to the binary Hamming scheme. Three types of self-dual codes are defined. For each type, the possible values $alpha,beta$ such that there exist a code $Csubseteq Z_2^alpha timesZ_4^beta$ are established. Moreover, the construction of a $add$-linear code for each type and possible pair $(alpha,beta)$ is given. Finally, the standard techniques of invariant theory are applied to describe the weight enumerators for each type.
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