Anna-Lena Trautmann, Joachim Rosenthal
We give a complete list of orbit codes that are generated by an irreducible cyclic group, i.e. an irreducible group having one generator. We derive some of the basic properties of these codes such as the cardinality and the minimum distance.
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Irreducible cyclic codes are an interesting type of codes and have applications in space communications. They have been studied for decades and a lot of progress has been made. The objectives of this paper are to survey and extend earlier results on the weight distributions of irreducible cyclic codes, present a divisibility theorem and develop bounds on the weights in irreducible cyclic codes.
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.
Up to a new invariant $μ(b)$, the complete $b$-symbol weight distribution of a particular kind of two-weight irreducible cyclic codes, was recently obtained by Zhu et al. [Des. Codes Cryptogr., 90 (2022) 1113-1125]. The purpose of this paper is to simplify and generalize the results of Zhu et al., and obtain the $b$-symbol weight distributions of all one-weight and two-weight semiprimitive irreducible cyclic codes.
In this paper, several families of irreducible constacyclic codes over finite fields and their duals are studied. The weight distributions of these irreducible constacyclic codes and the parameters of their duals are settled. Several families of irreducible constacyclic codes with a few weights and several families of optimal constacyclic codes are constructed. As by-products, a family of $[2n, (n-1)/2, d \geq 2(\sqrt{n}+1)]$ irreducible cyclic codes over $\gf(q)$ and a family of $[(q-1)n, (n-1)/2, d \geq (q-1)(\sqrt{n}+1)]$ irreducible cyclic codes over $\gf(q)$ are presented, where $n$ is a …
Recently, $b$-symbol codes are proposed to protect against $b$-symbol errors in $b$-symbol read channels. It is an interesting subject of study to consider the complete $b$-symbol weight distribution of cyclic codes since $b$-symbol metric is a generalization for Hamming metric. The complete $b$-symbol Hamming weight distribution of irreducible codes is known in only a few cases. In this paper, we give a complete $b$-symbol Hamming weight distribution of a class of irreducible codes with two nonzero $b$-symbol Hamming weights.
In coding theory, a very interesting problem (but at the same time, a very difficult one) is to determine the weight distribution of a given code. This problem is even more interesting for cyclic codes, and this is so, mainly because they possess a rich algebraic structure, which can be utilized in a variety of ways. For example, in Engineering Telecommunications, this structure is commonly employed to design very efficient coding and decoding algorithms. In this note, we are going to use a characterization of all semiprimitive two-weight irreducible cyclic codes over any finite field, recentl…
The generalized Hamming weight (GHW) $d_r(C)$ of linear codes $C$ is a natural generalization of the minimum Hamming distance $d(C)(=d_1(C))$ and has become one of important research objects in coding theory since Wei's originary work [23] in 1991. In this paper two general formulas on $d_r(C)$ for irreducible cyclic codes are presented by using Gauss sums and the weight hierarchy $\{d_1(C), d_2(C), \ldots, d_k(C)\}$ $(k=\dim C)$ are completely determined for several cases.
The $b$-symbol read channel is motivated by the limitations of the reading process in high density data storage systems. The corresponding new metric is a generalization of the Hamming metric known as the $b$-symbol weight metric and has become an important object in coding theory. In this paper, the general $b$-symbol weight enumerator formula for irreducible cyclic codes is presented by using the Gaussian period and a new invariant $\#U(b,j,N_1)$. The related $b$-symbol weight hierarchies $\{d_1(\C),d_2(\C),\ldots,d_K(\C)\}$ ($K=\dim(\C)$) are given for some cases. The shortened codes which …
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 $γg(x)=(x+d)^rg({A}(x))$, where $q=2^n$, $A=\left(\begin{array}{cc} a&b\\1&d\end{array}\right)\in PGL_2(\Bbb F_{q})$, $\mathrm{ord}(A)$ is a prime, $g(a)\ne 0$, and $0\ne γ\in \Bbb F_q$. And we give a complete characterization of irreducible polynomials $g(x)$ of degree $2s$ or $3s$ as above, where…
Recently, linear codes with a few weights were widely investigated due to their applications in secret sharing schemes and authentication schemes. In this letter, we present a class of $q$-ary linear codes derived from irreducible cyclic codes with $q$ a prime power. We use Gauss sums to represent its Hamming weights and obtain the bounds of its minimum Hamming distance. In some cases, we explicitly determine its weight distributions which generalize some known results. It is quite interesting that many new codes obtained in this letter are optimal according to some bounds on linear codes.
In this article, we show explicitly all possible weight enumerators for every irreducible cyclic code of length $n$ over a finite field $\mathbb F_q$, in the case which each prime divisor of $n$ is also a divisor of $q-1$.
Irreducible cyclic codes are one of the largest known classes of block codes which have been investigated for a long time. However, their weight distributions are known only for a few cases. In this paper, a class of irreducible cyclic codes are studied and their weight distributions are determined. Moreover, all codewords of some irreducible cyclic codes are obtained through programming in order to explain their distributions. The number of distinct nonzero weights in these codes dealt with in this paper varies among 1,2,3,6,8.
We explicitly determine the values of reduced cyclotomic periods of order $2^m$, $m\ge 4$, for finite fields of characteristic $p\equiv 3$ or $5\pmod{8}$. These evaluations are applied to obtain explicit factorizations of the corresponding reduced period polynomials. As another application, the weight distributions of certain irreducible cyclic codes are described.
The complete weight distribution of a family of irreducible cyclic codes of dimension two
本研究通过确定有限域上特定多项式的显式因式分解，无需计算指数和就得到了一类二维不可约循环码的完全重量分布，并据此构造出最优或几乎最优的系统认证码。
Constant dimension codes are subsets of the finite Grassmann variety. The study of these codes is a central topic in random linear network coding theory. Orbit codes represent a subclass of constant dimension codes. They are defined as orbits of a subgroup of the general linear group on the Grassmannian. This paper gives a complete characterization of orbit codes that are generated by an irreducible cyclic group, i.e. a group having one generator that has no non-trivial invariant subspace. We show how some of the basic properties of these codes, the cardinality and the minimum distance, can be…
The weight enumerators of irreducible cyclic codes of length $2^\ell$
本文利用不可约循环码与广义Paley图的谱关系，刻画长度为$2^\ell$的不可约循环码对应的广义Paley图类型，计算其谱后得到这类码重量枚举式的显式表达式。
For $q=p^m$ with $p$ prime and $k\mid q-1$, we consider the generalized Paley graph $Γ(k,q) = Cay(\mathbb{F}_q, R_k)$, with $R_k=\{ x^k : x \in \mathbb{F}_q^* \}$, and the irreducible $p$-ary cyclic code $\mathcal{C}(k,q) = \{(\textrm{Tr}_{q/p}(γω^{ik})_{i=0}^{n-1})\}_{γ\in \mathbb{F}_q}$, with $ω$ a primitive element of $\mathbb{F}_q$ and $n=\tfrac{q-1}{k}$. We first express the spectra of $Γ(k,q)$ in terms of Gaussian periods. Then, we show that the spectra of $Γ(k,q)$ and $\mathcal{C}(k,q)$ are mutually determined by each other if further $k\mid \tfrac{q-1}{p-1}$. We give $Spec(Γ(k,q))$ exp…
The cyclic codes with parity check polynomial the reciprocal of the characteristic polynomial of the Fibonacci recurrence over a prime finite field are shown to have either one weight or two weights. When these codes are irreducible cyclic we obtain many counterexamples to the conjectural classification of two-weight irreducible cyclic codes of Schmidt and White (2002). When they are reducible and projective their duals are uniformly packed.
Let \(\cU\) be the multiplicative group of order~\(n\) in the splitting field \(\bbF_{q^m}\) of \(x^n-1\) over the finite field \(\bbF_q\). Any map of the form \(x\rightarrow cx^t\) with \(c\in \cU\) and \(t=q^i\), \(0\leq i<m\), is \(\bbF_q\)-linear on~\(\bbF_{q^m}\) and fixes \(\cU\) set-wise; maps of this type will be called {\em standard\/}. Occasionally there are other, {\em non-standard\/} \(\bbF_q\)-linear maps on~\(\bbF_{q^m}\) fixing \(\cU\) set-wise, and in that case we say that the pair \((n, q)\) is {\em non-standard\/}. We show that an irreducible cyclic code of length~\(n\) over …
In network coding a constant dimension code consists of a set of k-dimensional subspaces of F_q^n. Orbit codes are constant dimension codes which are defined as orbits of a subgroup of the general linear group, acting on the set of all subspaces of F_q^n. If the acting group is cyclic, the corresponding orbit codes are called cyclic orbit codes. In this paper we give a classification of cyclic orbit codes and propose a decoding procedure for a particular subclass of cyclic orbit codes.
In network coding, a flag code is a set of sequences of nested subspaces of $\mathbb{F}_q^n$, being $\mathbb{F}_q$ the finite field with $q$ elements. Flag codes defined as orbits of a cyclic subgroup of the general linear group acting on flags of $\mathbb{F}_q^n$ are called cyclic orbit flag codes. Inspired by the ideas in arXiv:1403.1218, we determine the cardinality of a cyclic orbit flag code and provide bounds for its distance with the help of the largest subfield over which all the subspaces of a flag are vector spaces (the best friend of the flag). Special attention is paid to two speci…
The spectral relation between irreducible cyclic codes and generalized Paley graphs
本文建立了不可约循环码重量分布与广义Paley图谱之间的闭式联系，并显式给出了相关谱及重量枚举器。
On equidistant single-orbit cyclic and quasi-cyclic subspace codes
本文研究等距单轨道循环与准循环子空间码，借助循环差集证明仅存在平凡等距单轨道循环子空间码，并进一步探究了向日葵型等距单轨道准循环子空间码。
Cyclic orbit codes are constant dimension subspace codes that arise as the orbit of a cyclic subgroup of the general linear group acting on subspaces in the given ambient space. With the aid of the largest subfield over which the given subspace is a vector space, the cardinality of the orbit code can be determined, and estimates for its distance can be found. This subfield is closely related to the stabilizer of the generating subspace. Finally, with a linkage construction larger, and longer, constant dimension codes can be derived from cyclic orbit codes without compromising the distance.
In this article, we count the quantity of minimal cyclic codes of length $n$ and dimension $k$ over a finite field $\mathbb F_q$, in the case when the prime factors of $n$ satisfy a special condition. This problem is equivalent to count the quantity of irreducible factors of $x^n-1\in \mathbb F_q[x]$ of degree $k$.
Cyclic orbit codes are a family of constant dimension codes used for random network coding. We investigate the Plücker embedding of these codes and show how to efficiently compute the Grassmann coordinates of the code words.
We present a family of reducible cyclic codes constructed as the direct sum of two different semiprimitive two-weight irreducible cyclic codes. This family generalizes the class of reducible cyclic codes that was reported in the main result of B. Wang, {\em et al.} \cite{once}. Moreover, despite of what was stated therein, we show that, at least for the codes studied here, it is still possible to compute the frequencies of their weight distributions through the cyclotomic numbers in a very easy way.
Orbit codes are a family of codes employable for communications on a random linear network coding channel. The paper focuses on the classification of these codes. We start by classifying the conjugacy classes of cyclic subgroups of the general linear group. As a result, we are able to focus the study of cyclic orbit codes to a restricted family of them.
We give necessary conditions for a two-weight projective cyclic code to be the direct sum of two one-weight irreducible cyclic subcodes of the same dimension, following the work of Wolfmann and Vega. This confirms Vega's conjecture that all the two-weight cyclic codes of this type are the known ones in the projective case.
Flag codes that are orbits of a cyclic subgroup of the general linear group acting on flags of a vector space over a finite field, are called cyclic orbit flag codes. In this paper we present a new contribution to the study of such codes started in arXiv:2102.00867, by focusing this time on the generating flag. More precisely, we examine those ones whose generating flag has at least one subfield among its subspaces. In this situation, two important families arise: the already known Galois flag codes, in case we have just fields, or the generalized Galois flag codes in other case. We investigat…
Spread codes and cyclic orbit codes are special families of constant dimension subspace codes. These codes have been well-studied for their error correction capability, transmission rate and decoding methods, but the question of how to encode and retrieve messages has not been investigated. In this work we show how a message set of consecutive integers can be encoded and retrieved for these two code families.
Relative generalized Hamming weights (RGHWs) of a linear code respect to a linear subcode determine the security of the linear ramp secret sharing scheme based on the code. They can be used to express the information leakage of the secret when some keepers of shares are corrupted. Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems. In this paper, we investigate the RGHWs of cyclic codes with two nonzeros respect to any of its irreducible cyclic subcodes. Applying the method in the paper [arxiv.org/abs/1410.2702], we give two for…
Upper and lower bounds on the largest number of weights in a cyclic code of given length, dimension and alphabet are given. An application to irreducible cyclic codes is considered. Sharper upper bounds are given for the special cyclic codes (called here strongly cyclic), {whose nonzero codewords have period equal to the length of the code}. Asymptotics are derived on the function $Γ(k,q),$ {that is defined as} the largest number of nonzero weights a cyclic code of dimension $k$ over $\F_q$ can have, and an algorithm to compute it is sketched. The nonzero weights in some infinite families of R…
The symbol-pair codes over finite fields have been raised for symbol-pair read channels and motivated by application of high-density data storage technologies [1, 2]. Their generalization is the code for b-symbol read channels (b > 2). Many MDS codes for b-symbol read channels have been constructed which meet the Singleton-like bound ([3, 4, 10] for b = 2 and [11] for b > 2). In this paper we show the Plotkin-like bound and present a construction on irreducible cyclic codes and constacyclic codes meeting the Plotkin-like bound.
Spread Codes from Abelian non-cyclic groups
本文提出基于Abelian非循环群作用构造有限域F_q^n铺砌的方法，并利用该群构建最大尺寸轨道码族以实现具有轨道结构的铺砌。
Ovoids in $\PG(3, q)$ have been an interesting topic in coding theory, combinatorics, and finite geometry for a long time. So far only two families are known. The first is the elliptic quadratics and the second is the Tits ovoids. In this article, we present a family of ovoids in $\PG(3, 2^m)$ for all $m$ which are from a family of irreducible cyclic codes.
We study orbit codes in the field extension ${\mathbb F}_{q^n}$. First we show that the automorphism group of a cyclic orbit code is contained in the normalizer of the Singer subgroup if the orbit is generated by a subspace that is not contained in a proper subfield of ${\mathbb F}_{q^n}$. We then generalize to orbits under the normalizer of the Singer subgroup. In that situation some exceptional cases arise and some open cases remain. Finally we characterize linear isometries between such codes.
In this paper, we study the codes over the matrix ring over $\mathbb{Z}_4$, which is perhaps the first time the ring structure $M_2(\mathbb{Z}_4)$ is considered as a code alphabet. This ring is isomorphic to $\mathbb{Z}_4[w]+U\mathbb{Z}_4[w]$, where $w$ is a root of the irreducible polynomial $x^2+x+1 \in \mathbb{Z}_2[x]$ and $U\equiv$ ${11}\choose{11}$. We first discuss the structure of the ring $M_2(\mathbb{Z}_4)$ and then focus on algebraic structure of cyclic codes and self-dual cyclic codes over $M_2(\mathbb{Z}_4)$. We obtain the generators of the cyclic codes and their dual codes. Few ex…
Full weight spectrum one-orbit cyclic subspace codes
本文研究循环子空间码框架下的全重量谱单轨道循环子空间码，利用数论结果与组合论证，给出了该类码的完整分类。
Quasi-optimal cyclic orbit codes
本文研究循环轨道码的等价不变量与拟最优性，通过射影直线线性集获得参数新界，并证明任意特征有限域偶数维空间中拟最优码的一般存在性定理。
Using multi-orbit cyclic subspace codes for constructing optical orthogonal codes
本文提出利用多轨道循环子空间码和有限域扩张的乘法结构构造大型光正交码，不同于以往基于组合与加法结构的方法，从而获得了具有不同参数的新光正交码类。
Cyclic subspace codes gained a lot of attention especially because they may be used in random network coding for correction of errors and erasures. Roth, Raviv and Tamo in 2018 established a connection between cyclic subspace codes (with certain parameters) and Sidon spaces. These latter objects were introduced by Bachoc, Serra and Zémor in 2017 in relation with the linear analogue of Vosper's Theorem. This connection allowed Roth, Raviv and Tamo to construct large classes of cyclic subspace codes with one or more orbits. In this paper we will investigate cyclic subspace codes associated to a …
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 dist…
Cyclic codes are a subclass of linear codes and have wide applications in data storage systems, communication systems and consumer electronics due to their efficient encoding and decoding algorithms. Let $p\ge 5$ be an odd prime and $m$ be a positive integer. Let $\mathcal{C}_{(1,e,s)}$ denote the $p$-ary cyclic code with three nonzeros $α$, $α^e$, and $α^s$, where $α$ is a generator of ${\mathbb F}_{p^m}^*$, $s=\frac{p^m-1}{2}$, and $2\le e\le p^m-2$. In this paper, we present four classes of optimal $p$-ary cyclic codes $\mathcal{C}_{(1,e,s)}$ with parameters $[p^m-1,p^m-2m-2,4]$ by analyzin…
On one-orbit cyclic subspace codes of $\mathcal{G}_q(n,3)$
本文研究三维单轨道循环子空间码，将其分为三个族，并重点分析后两个族中不等价码的性质，以推进循环子空间码的分类研究。
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