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Register a new userA Note on Cyclic Codes from APN Functions
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Chunming Tang
Publication date
2013
fields
Informatics Engineering
and research's language is
English
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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}.
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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.
In 2020, Budaghyan, Helleseth and Kaleyski [IEEE TIT 66(11): 7081-7087, 2020] considered an infinite family of quadrinomials over $mathbb{F}_{2^{n}}$ of the form $x^3+a(x^{2^s+1})^{2^k}+bx^{3cdot 2^m}+c(x^{2^{s+m}+2^m})^{2^k}$, where $n=2m$ with $m$ odd. They proved that such kind of quadrinomials can provide new almost perfect nonlinear (APN) functions when $gcd(3,m)=1$, $ k=0 $, and $(s,a,b,c)=(m-2,omega, omega^2,1)$ or $((m-2)^{-1}~{rm mod}~n,omega, omega^2,1)$ in which $omegainmathbb{F}_4setminus mathbb{F}_2$. By taking $a=omega$ and $b=c=omega^2$, we observe that such kind of quadrinomials can be rewritten as $a {rm Tr}^{n}_{m}(bx^3)+a^q{rm Tr}^{n}_{m}(cx^{2^s+1})$, where $q=2^m$ and $ {rm Tr}^n_{m}(x)=x+x^{2^m} $ for $ n=2m$. Inspired by the quadrinomials and our observation, in this paper we study a class of functions with the form $f(x)=a{rm Tr}^{n}_{m}(F(x))+a^q{rm Tr}^{n}_{m}(G(x))$ and determine the APN-ness of this new kind of functions, where $a in mathbb{F}_{2^n} $ such that $ a+a^q eq 0$, and both $F$ and $G$ are quadratic functions over $mathbb{F}_{2^n}$. We first obtain a characterization of the conditions for $f(x)$ such that $f(x) $ is an APN function. With the help of this characterization, we obtain an infinite family of APN functions for $ n=2m $ with $m$ being an odd positive integer: $ f(x)=a{rm Tr}^{n}_{m}(bx^3)+a^q{rm Tr}^{n}_{m}(b^3x^9) $, where $ ain mathbb{F}_{2^n}$ such that $ a+a^q eq 0 $ and $ b $ is a non-cube in $ mathbb{F}_{2^n} $.
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.
This short note revisits the problem of designing secure minimum storage regenerating (MSR) codes for distributed storage systems. A secure MSR code ensures that a distributed storage system does not reveal the stored information to a passive eavesdropper. The eavesdropper is assumed to have access to the content stored on $ell_1$ number of storage nodes in the system and the data downloaded during the bandwidth efficient repair of an additional $ell_2$ number of storage nodes. This note combines the Gabidulin codes based precoding [18] and a new construction of MSR codes (without security requirements) by Ye and Barg [27] in order to obtain secure MSR codes. Such optimal secure MSR codes were previously known in the setting where the eavesdropper was only allowed to observe the repair of $ell_2$ nodes among a specific subset of $k$ nodes [7, 18]. The secure coding scheme presented in this note allows the eavesdropper to observe repair of any $ell_2$ ouf of $n$ nodes in the system and characterizes the secrecy capacity of linear repairable MSR codes.
A linear code is called an MDS self-dual code if it is both an MDS code and a self-dual code with respect to the Euclidean inner product. The parameters of such codes are completely determined by the code length. In this paper, we consider new constructions of MDS self-dual codes via generalized Reed-Solomon (GRS) codes and their extended codes. The critical idea of our constructions is to choose suitable evaluation points such that the corresponding (extended) GRS codes are self-dual. The evaluation set of our constructions is consists of a subgroup of finite fields and its cosets in a bigger subgroup. Four new families of MDS self-dual codes are obtained and they have better parameters than previous results in certain region. Moreover, by the Mobius action over finite fields, we give a systematic way to construct self-dual GRS codes with different evaluation points provided any known self-dual GRS codes. Specially, we prove that all the self-dual extended GRS codes over $mathbb{F}_{q}$ with length $n< q+1$ can be constructed from GRS codes with the same parameters.
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