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Register a new userA Note on Weight Distributions of Irreducible Cyclic Codes
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Chunming Tang
Publication date
2012
fields
Informatics Engineering
and research's language is
English
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Usually, it is difficult to determine the weight distribution of an irreducible cyclic code. In this paper, we discuss the case when an irreducible cyclic code has the maximal number of distinct nonzero weights and give a necessary and sufficient condition. In this case, we also obtain a divisible property for the weight of a codeword. Further, we present a necessary and sufficient condition for an irreducible cyclic code with only one nonzero weight. Finally, we determine the weight distribution of an irreducible cyclic code for some cases.
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Cyclic codes with two zeros and their dual codes as a practically and theoretically interesting class of linear codes, have been studied for many years. However, the weight distributions of cyclic codes are difficult to determine. From elliptic curves, this paper determines the weight distributions of dual codes of cyclic codes with two zeros for a few more cases.
Cyclic codes, as linear block error-correcting codes in coding theory, play a vital role and have wide applications. Ding in cite{D} constructed a number of classes of cyclic codes from almost perfect nonlinear (APN) functions and planar functions over finite fields and presented ten open problems on cyclic codes from highly nonlinear functions. In this paper, we consider two open problems involving the inverse APN functions $f(x)=x^{q^m-2}$ and the Dobbertin APN function $f(x)=x^{2^{4i}+2^{3i}+2^{2i}+2^{i}-1}$. From the calculation of linear spans and the minimal polynomials of two sequences generated by these two classes of APN functions, the dimensions of the corresponding cyclic codes are determined and lower bounds on the minimum weight of these cyclic codes are presented. Actually, we present a framework for the minimal polynomial and linear span of the sequence $s^{infty}$ defined by $s_t=Tr((1+alpha^t)^e)$, where $alpha$ is a primitive element in $GF(q)$. These techniques can also be applied into other open problems in cite{D}.
Goppa codes are particularly appealing for cryptographic applications. Every improvement of our knowledge of Goppa codes is of particular interest. In this paper, we present a sufficient and necessary condition for an irreducible monic polynomial $g(x)$ of degree $r$ over $mathbb{F}_{q}$ satisfying $gamma g(x)=(x+d)^rg({A}(x))$, where $q=2^n$, $A=left(begin{array}{cc} a&b1&dend{array}right)in PGL_2(Bbb F_{q})$, $mathrm{ord}(A)$ is a prime, $g(a) e 0$, and $0 e gammain Bbb F_q$. And we give a complete characterization of irreducible polynomials $g(x)$ of degree $2s$ or $3s$ as above, where $s$ is a positive integer. Moreover, we construct some binary irreducible quasi-cyclic parity-check subcodes of Goppa codes and extended Goppa codes.
The problem of identifying whether the family of cyclic codes is asymptotically good or not is a long-standing open problem in the field of coding theory. It is known in the literature that some families of cyclic codes such as BCH codes and Reed-Solomon codes are asymptotically bad, however in general the answer to this question is not known. A recent result by Nelson and Van Zwam shows that, all linear codes can be obtained by a sequence of puncturing and/or shortening of a collection of asymptotically good codes~cite{Nelson_2015}. In this paper, we prove that any linear code can be obtained by a sequence of puncturing and/or shortening of some cyclic code. Therefore the result that all codes can be obtained by shortening and/or puncturing cyclic codes leaves the possibility open that cyclic codes are asymptotically good.
The distance distribution of a code is the vector whose $i^text{th}$ entry is the number of pairs of codewords with distance $i$. We investigate the structure of the distance distribution for cyclic orbit codes, which are subspace codes generated by the action of $mathbb{F}_{q^n}^*$ on an $mathbb{F}_q$-subspace $U$ of $mathbb{F}_{q^n}$. We show that for optimal full-length orbit codes the distance distribution depends only on $q,,n$, and the dimension of $U$. For full-length orbit codes with lower minimum distance, we provide partial results towards a characterization of the distance distribution, especially in the case that any two codewords intersect in a space of dimension at most 2. Finally, we briefly address the distance distribution of a union of optimal full-length orbit codes.
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