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Register a new userOn $Z_pZ_{p^k}$-additive codes and their duality
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Added by
Denis Krotov
Publication date
2018
fields
Informatics Engineering
and research's language is
English
Authors
Minjia Shi Anhuin University
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In this paper, two different Gray-like maps from $Z_p^alphatimes Z_{p^k}^beta$, where $p$ is prime, to $Z_p^n$, $n={alpha+beta p^{k-1}}$, denoted by $phi$ and $Phi$, respectively, are presented. We have determined the connection between the weight enumerators among the image codes under these two mappings. We show that if $C$ is a $Z_p Z_{p^k}$-additive code, and $C^bot$ is its dual, then the weight enumerators of the image $p$-ary codes $phi(C)$ and $Phi(C^bot)$ are formally dual. This is a partial generalization of [On $Z_{2^k}$-dual binary codes, arXiv:math/0509325], and the result is generalized to odd characteristic $p$ and mixed alphabet. Additionally, a construction of $1$-perfect additive codes in the mixed $Z_p Z_{p^2} ... Z_{p^k}$ alphabet is given.
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The well known Plotkin construction is, in the current paper, generalized and used to yield new families of Z2Z4-additive codes, whose length, dimension as well as minimum distance are studied. These new constructions enable us to obtain families of Z2Z4-additive codes such that, under the Gray map, the corresponding binary codes have the same parameters and properties as the usual binary linear Reed-Muller codes. Moreover, the first family is the usual binary linear Reed-Muller family.
The Doob graph $D(m,n)$ is the Cartesian product of $m>0$ copies of the Shrikhande graph and $n$ copies of the complete graph of order $4$. Naturally, $D(m,n)$ can be represented as a Cayley graph on the additive group $(Z_4^2)^m times (Z_2^2)^{n} times Z_4^{n}$, where $n+n=n$. A set of vertices of $D(m,n)$ is called an additive code if it forms a subgroup of this group. We construct a $3$-parameter class of additive perfect codes in Doob graphs and show that the known necessary conditions of the existence of additive $1$-perfect codes in $D(m,n+n)$ are sufficient. Additionally, two quasi-cyclic additive $1$-perfect codes are constructed in $D(155,0+31)$ and $D(2667,0+127)$.
We consider DNA codes based on the nearest-neighbor (stem) similarity model which adequately reflects the hybridization potential of two DNA sequences. Our aim is to present a survey of bounds on the rate of DNA codes with respect to a thermodynamically motivated similarity measure called an additive stem similarity. These results yield a method to analyze and compare known samples of the nearest neighbor thermodynamic weights associated to stacked pairs that occurred in DNA secondary structures.
We obtain a characterization on self-orthogonality for a given binary linear code in terms of the number of column vectors in its generator matrix, which extends the result of Bouyukliev et al. (2006). As an application, we give an algorithmic method to embed a given binary $k$-dimensional linear code $mathcal{C}$ ($k = 2,3,4$) into a self-orthogonal code of the shortest length which has the same dimension $k$ and minimum distance $d ge d(mathcal{C})$. For $k > 4$, we suggest a recursive method to embed a $k$-dimensional linear code to a self-orthogonal code. We also give new explicit formulas for the minimum distances of optimal self-orthogonal codes for any length $n$ with dimension 4 and any length $n otequiv 6,13,14,21,22,28,29 pmod{31}$ with dimension 5. We determine the exact optimal minimum distances of $[n,4]$ self-orthogonal codes which were left open by Li-Xu-Zhao (2008) when $n equiv 0,3,4,5,10,11,12 pmod{15}$. Then, using MAGMA, we observe that our embedding sends an optimal linear code to an optimal self-orthogonal code.
Let $mathbb{F}_{q}$ be the finite field of $q$ elements and let $D_{2n}=langle x,ymid x^n=1, y^2=1, yxy=x^{n-1}rangle$ be the dihedral group of order $n$. Left ideals of the group algebra $mathbb{F}_{q}[D_{2n}]$ are known as left dihedral codes over $mathbb{F}_{q}$ of length $2n$, and abbreviated as left $D_{2n}$-codes. Let ${rm gcd}(n,q)=1$. In this paper, we give an explicit representation for the Euclidean hull of every left $D_{2n}$-code over $mathbb{F}_{q}$. On this basis, we determine all distinct Euclidean LCD codes and Euclidean self-orthogonal codes which are left $D_{2n}$-codes over $mathbb{F}_{q}$. In particular, we provide an explicit representation and a precise enumeration for these two subclasses of left $D_{2n}$-codes and self-dual left $D_{2n}$-codes, respectively. Moreover, we give a direct and simple method for determining the encoder (generator matrix) of any left $D_{2n}$-code over $mathbb{F}_{q}$, and present several numerical examples to illustrative our applications.
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