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Register a new userOn Binary Cyclic Codes with Five Nonzero Weights
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Added by
Luo Jinquan
Publication date
2009
fields
Informatics Engineering
and research's language is
English
Authors
Jinquan Luo
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Let $q=2^n$, $0leq kleq n-1$, $n/gcd(n,k)$ be odd and $k eq n/3, 2n/3$. In this paper the value distribution of following exponential sums [sumlimits_{xin bF_q}(-1)^{mathrm{Tr}_1^n(alpha x^{2^{2k}+1}+beta x^{2^k+1}+ga x)}quad(alpha,beta,gain bF_{q})] is determined. As an application, the weight distribution of the binary cyclic code $cC$, with parity-check polynomial $h_1(x)h_2(x)h_3(x)$ where $h_1(x)$, $h_2(x)$ and $h_3(x)$ are the minimal polynomials of $pi^{-1}$, $pi^{-(2^k+1)}$ and $pi^{-(2^{2k}+1)}$ respectively for a primitive element $pi$ of $bF_q$, is also determined.
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In this paper, we make some progress towards a well-known conjecture on the minimum weights of binary cyclic codes with two primitive nonzeros. We also determine the Walsh spectrum of $Tr(x^d)$ over $F_{2^{m}}$ in the case where $m=2t$, $d=3+2^{t+1}$ and $gcd(d, 2^{m}-1)=1$.
The Doob scheme $D(m,n+n)$ is a metric association scheme defined on $E_4^m times F_4^{n}times Z_4^{n}$, where $E_4=GR(4^2)$ or, alternatively, on $Z_4^{2m} times Z_2^{2n} times Z_4^{n}$. We prove the MacWilliams identities connecting the weight distributions of a linear or additive code and its dual. In particular, for each case, we determine the dual scheme, on the same set but with different metric, such that the weight distribution of an additive code $C$ in the Doob scheme $D(m,n+n)$ is related by the MacWilliams identities with the weight distribution of the dual code $C^perp$ in the dual scheme. We note that in the case of a linear code $C$ in $E_4^m times F_4^{n}$, the weight distributions of $C$ and $C^perp$ in the same scheme are also connected.
For any integer $rho geq 1$ and for any prime power q, the explicit construction of a infinite family of completely regular (and completely transitive) q-ary codes with d=3 and with covering radius $rho$ is given. The intersection array is also computed. Under the same conditions, the explicit construction of an infinite family of q-ary uniformly packed codes (in the wide sense) with covering radius $rho$, which are not completely regular, is also given. In both constructions the Kronecker product is the basic tool that has been used.
One of the most important and challenging problems in coding theory is to construct codes with best possible parameters and properties. The class of quasi-cyclic (QC) codes is known to be fertile to produce such codes. Focusing on QC codes over the binary field, we have found 113 binary QC codes that are new among the class of QC codes using an implementation of a fast cyclic partitioning algorithm and the highly effective ASR algorithm. Moreover, these codes have the following additional properties: a) they have the same parameters as best known linear codes, and b) many of the have additional desired properties such as being reversible, LCD, self-orthogonal or dual-containing. Additionally, we present an algorithm for the generation of new codes from QC codes using ConstructionX, and introduce 35 new record breaking linear codes produced from this method.
In this paper, we apply two-to-one functions over $mathbb{F}_{2^n}$ in two generic constructions of binary linear codes. We consider two-to-one functions in two forms: (1) generalized quadratic functions; and (2) $left(x^{2^t}+xright)^e$ with $gcd(t, n)=1$ and $gcdleft(e, 2^n-1right)=1$. Based on the study of the Walsh transforms of those functions or their related-ones, we present many classes of linear codes with few nonzero weights, including one weight, three weights, four weights and five weights. The weight distributions of the proposed codes with one weight and with three weights are determined. In addition, we discuss the minimum distance of the dual of the constructed codes and show that some of them achieve the sphere packing bound. { Moreover, several examples show that some of our codes are optimal and some have the best known parameters.}
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