On Some Classes of $\mathbb{Z}_{2}\mathbb{Z}_{4}-$Linear Codes and their Covering Radius
K. Chatouh, K. Guenda, T. A. Gulliver, L. Noui
本文定义了$α$和$β$型的$\mathbb{Z}_{2}\mathbb{Z}_{4}-$单纯码和MacDonald码，并给出了这类码的覆盖半径。
In this paper we define $\mathbb{Z}_{2}\mathbb{Z}_{4}-$Simplex and MacDonald Codes of type $α$ and $β$ and we give the covering radius of these codes.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
This paper gives lower and upper bounds on the covering radius of codes over $\mathbb{Z}_{p^2}$ with respect to Lee distance. We also determine the covering radius of various Repetition codes over $\mathbb{Z}_{p^2}.$
The Second Generalized Covering Radius of Binary Primitive Double-Error-Correcting BCH Codes
该研究完全确定了二元本原双纠错BCH码的第二覆盖半径，并给出了纠错数大于2的二元本原BCH码第二覆盖半径的下界。
It was proved by J. Schatz that the covering radius of the second order Reed--Muller code $RM(2, 6)$ is 18 (IEEE Trans Inf Theory 27: 529--530, 1985). However, the covering radius of $RM(2,7)$ has been an open problem for many years. In this paper, we prove that the covering radius of $RM(2,7)$ is 40, which is the same as the covering radius of $RM(2,7)$ in $RM(3,7)$. As a corollary, we also find new upper bounds for $RM(2,n)$, $n=8,9,10$.
On the Covering Radius of the Second Order Reed-Muller Code of Length 128
1981年，Schatz证明了二元Reed-Muller码$RM(2,6)$的覆盖半径为18。对于$RM(2,7)$，我们只知道它的覆盖半径在40到44之间。本文证明了二元Reed-Muller码$RM(2,7)$的覆盖半径不超过42。此外，我们给出了一个充分必要条件，使7元布尔函数达到二阶非线性42。
On the burst-covering radius of binary cyclic codes
本文研究了突发覆盖码的定义与性质，推导了循环码突发覆盖半径的更紧界限，并提出了高效的突发覆盖算法及基于突发覆盖半径的循环码临界指数界。
We classify all linear completely regular codes which have covering radius $ρ= 2$ and whose dual are antipodal. For this, we firstly show several properties of such dual codes, which are two-weight codes.
On a family of binary completely transitive codes with growing covering radius
构造了一类新的二元线性完全传递（因此也是完全正则）码。这些码的覆盖半径随着码的长度而增长。特别地，对于任意整数r > 1，存在两个d=3的码，分别覆盖半径r和长度2 r（4 r-1）和（2 r +1）（4 r +1）。这些新的完全传递代码诱导，陪集图，一个家庭的距离传递图的增长直径。
On linear completely regular codes with covering radius $ρ=1$. Construction and classification
具有覆盖半径$ρ=1$的完全正则码必须具有最小距离$d\leq 3$。对于$d=3$，这样的码是完美的，并且它们的参数是公知的。本文研究了$d=1$和$d=2$的情形，并在码是线性的情况下完全刻画了这两种情形.此外，还证明了所有这些码都是完全传递的。
In this paper we present a family of $q$-ary nonlinear quasi-perfect codes with covering radius 2. The codes have length $n = q^m$ and size $ M = q^{n - m - 1}$ where $q$ is a prime power, $q \geq 3$, $m$ is an integer, $m \geq 2$. We prove that there are more than $q^{q^{cn}}$ nonequivalent such codes of length $n$, for all sufficiently large $n$ and a constant $c = \frac{1}{q} - \varepsilon$.
Rationality and computability of the covering radius for sofic shifts
研究证明了原始sofic shifts的覆盖半径是有理数，并提出从标记图呈现计算覆盖半径的算法。
On the covering radius of first order generalized Reed-Muller codes
本文将Helleseth，Klove和Mykkelvit证明的关于线性码覆盖半径的一个定理推广到任意q。然后确定了一阶广义Reed-Muller码在二阶广义Reed-Muller码中的覆盖半径。利用这些结果，我们能够给出一阶广义Reed-Muller码的覆盖半径的界。最后，使用Magma，我们得到了q=3的一些改进。
On the Covering Radius of Some Modular Codes
本文给出了$\Z_{2^s}$上码关于齐次距离的覆盖半径的上下界。我们还确定了各种重复码、单纯形码（$α$型和$β$型）及其对偶码的覆盖半径，并给出了$\Z_4$上这两类MacDonald码的覆盖半径的界.
Covering Sequences and Covering-Sequences Codes
研究如何利用汉明码构造覆盖序列及覆盖序列码，针对小半径情况构建短长度序列和相关码，还构造了长度渐近最优的小半径序列及较大半径的有趣码。
We study covering codes of permutations with the $\ell_\infty$-metric. We provide a general code construction, which uses smaller building-block codes. We study cyclic transitive groups as building blocks, determining their exact covering radius, and showing linear-time algorithms for finding a covering codeword. We also bound the covering radius of relabeled cyclic transitive groups under conjugation.
We present methods for computing the distance from a Boolean polynomial on $m$ variables of degree $m-3$ (i.e., a member of the Reed-Muller code $RM(m-3,m)$) to the space of lower-degree polynomials ($RM(m-4,m)$). The methods give verifiable certificates for both the lower and upper bounds on this distance. By applying these methods to representative lists of polynomials, we show that the covering radius of $RM(4,8)$ in $RM(5,8)$ is 26 and the covering radius of $RM(5,9)$ in $RM(6,9)$ is between 28 and 32 inclusive, and we get improved lower bounds for higher~$m$. We also apply our methods to …
A notion of $t$-designs in the symmetric group on $n$ letters was introduced by Godsil in 1988. In particular $t$-transitive sets of permutations form a $t$-design. We derive upper bounds on the covering radius of these designs, as a function of $n$ and $t$ and in terms of the largest zeros of Charlier polynomials.
The Covering Radius of the Reed--Muller Code RM(1,9) is 242
本文解决了密码学与编码理论中长期存在的Reed-Muller码RM(1,9)覆盖半径的开放问题，证明其覆盖半径为242，通过三次布尔函数及相关不等式推导完成非线性度排除。
Tietäväinen's upper and lower bounds assert that for block-length-$n$ linear codes with dual distance $d$, the covering radius $R$ is at most $\frac{n}{2}-(\frac{1}{2}-o(1))\sqrt{dn}$ and typically at least $\frac{n}{2}-Θ(\sqrt{dn\log{\frac{n}{d}}})$. The gap between those bounds on $R -\frac{n}{2}$ is an $Θ(\sqrt{\log{\frac{n}{d}}})$ factor related to the gap between the worst covering radius given $d$ and the sphere-covering bound. Our focus in this paper is on the case when $d = o(n)$, i.e., when the code size is subexponential and the gap is $w(1)$. We show that up to a constant, the gap c…
The list-decodable code has been an active topic in theoretical computer science.There are general results about the list-decodability to the Johnson radius and the list-decoding capacity theorem. In this paper we show that rates, list-decodable radius and list sizes are closely related to the classical topic of covering codes. We prove new general simple but strong upper bounds for list-decodable codes in general finite metric spaces based on various covering codes. The general covering code upper bounds can be applied to the case that the volumes of the balls depend on the centers, not only …
We introduce a novel framework for implementing error-correction in constrained systems. The main idea of our scheme, called Quantized-Constraint Concatenation (QCC), is to employ a process of embedding the codewords of an error-correcting code in a constrained system as a (noisy, irreversible) quantization process. This is in contrast to traditional methods, such as concatenation and reverse concatenation, where the encoding into the constrained system is reversible. The possible number of channel errors QCC is capable of correcting is linear in the block length $n$, improving upon the $O(\sq…
Auxiliary Codes and the Generalized Packing-Covering Conjecture
本文研究广义打包-覆盖猜想，证明任意有限域上冗余度至多十四的线性码均满足该猜想，并利用辅助码准则证明广义汉明重量界，同时分析二元本原BCH码的打包半径与覆盖半径关系。
Covering Radius and Deep Holes of Hyperderivative Reed-Solomon Codes under the NRT Metric
本文研究NRT度量下HRS码的覆盖半径与深孔，确定其覆盖半径，通过MDS扩展等刻画深孔，关联深孔计数与超平面排列特征多项式，给出深孔构造、提出猜想并推导特殊情形的精确计数公式。
The Exact Second Generalized Covering Radius of Binary Primitive Triple-Error-Correcting BCH Codes
本研究精确确定了长度为$2^m-1$的二元本原三重纠错BCH码$\text{BCH}(3,m)$的第二广义覆盖半径为8（对所有$m\not\neq5$），匹配已知下界，解决了该码族的该参数确定问题。
Motivated by an application to database linear querying, such as private information-retrieval protocols, we suggest a fundamental property of linear codes -- the generalized covering radius. The generalized covering-radius hierarchy of a linear code characterizes the trade-off between storage amount, latency, and access complexity, in such database systems. Several equivalent definitions are provided, showing this as a combinatorial, geometric, and algebraic notion. We derive bounds on the code parameters in relation with the generalized covering radii, study the effect of simple code operati…
On Octonary Codes and their Covering Radii
本文介绍了一种新的八进制码的约简码和扭码，并确定了它们的基本性质。这些结果对$\mathbb{Z}_8$上自正交码和自对偶码的分类有一定的参考价值。我们还重点研究了八进制码的覆盖半径问题。特别地，我们确定了$\mathbb{Z}_8$上的几类重复码、$α$型单纯形码和$β$型单纯形码及其子码、MacDonald码和Reed-Muller码的覆盖半径的上下界.
On Nearly-Perfect Covering Codes Beyond Radius One
本文研究二元近完备覆盖码，修正原Van Wee界以纳入最小距离，确定R=2、3的此类码等价于构造的码，且R≥3时此类码数量有限。
We present a generalization of Walsh-Hadamard transform that is suitable for applications in Coding Theory, especially for computation of the weight distribution and the covering radius of a linear code over a finite field. The transform used in our research, is a modification of Vilenkin-Chrestenson transform. Instead of using all the vectors in the considered space, we take a maximal set of nonproportional vectors, which reduces the computational complexity.
Covering radius of rank-metric codes via covering lifts and clubs
该研究提出基于生成矩阵的几何方法求$\text{F}_{q^m}$线性秩距离码的覆盖半径，针对1维码等情形给出判据、等价解释及构造，发现相同参数的1维秩距离码覆盖半径可不同。
Determining the covering radius of all generalized Zetterberg codes in odd characteristic
本文针对奇特征有限域上的广义Zetterberg码，利用有限域算术与代数曲线技术解决了剩余情形$q_0^s \equiv 7 \pmod{8}$下的覆盖半径公开问题，并推广到扭曲半广义Zetterberg码，得到若干准完美码。
A Generalized Covering Algorithm for Chained Codes
本文提出一种广义覆盖算法，将普通覆盖半径的已知界推广到满足链条件的码，并针对Reed-Muller码提供高效生成矩阵构造，实现更快的广义覆盖求解。
Efficient Covering Using Reed--Solomon Codes
本文首次提出利用Berlekamp-Welch和Guruswami-Sudan译码算法高效求解Reed-Solomon码覆盖问题，并通过理论与数值结果证明接近列表译码半径的汉明球能覆盖大部分空间。
Saturating systems and the rank covering radius
本文引入秩饱和系统概念并建立其与秩度量码覆盖半径的对应关系，研究了求解最小维数 $s_{q^m/q}(k,ρ)$ 的秩度量覆盖问题，给出了该值的上下界、特定参数求值及基于几何的系统构造。
We prove the covering radius of the third-order Reed-Muller code RM(3,7) is 20, which was previously known to be between 20 and 23 (inclusive). The covering radius of RM(3, 7) is the maximum third-order nonlinearity among all 7-variable Boolean functions. It was known that there exist 7-variable Boolean functions with third-order nonlinearity 20. We prove the third-order nonlinearity cannot achieve 21. According to the classification of the quotient space of RM(6,6)/RM(3,6), we classify all 7-variable Boolean functions into 66 types. Firstly, we prove 62 types (among 66) cannot have third-orde…
Covering in Hamming and Grassmann Spaces: New Bounds and Reed--Solomon-Based Constructions
本文研究了汉明和格拉斯曼空间中的覆盖问题，提出基于里德-所罗门码的新构造，并展示了其在平均覆盖性能上的优势。
On the Generalized Packing and Covering Radii of Codes
本文证明了码的广义填充半径不超过同阶广义覆盖半径的猜想，对二阶半径、码率至多3/5的所有阶以及任意码率下足够长的码均成立。
The multicovering radius problem for some types of discrete structures
覆盖半径问题（英语：Covering radius problem）是编码理论中的一个问题，涉及寻找最小半径$r$，使得给定一个作为底层度量空间子集的码，其码字上半径为$r$的球覆盖整个度量空间。Klapper引入了一个代码参数，称为多重覆盖半径，它是覆盖半径的推广。本文引入了置换码的多重覆盖半径的一个类似形式（参见Keevash和Ku，2006年）和完美匹配的代码（cf. Aw和Ku，2012年）。我们应用概率工具给出了这些码的多重覆盖半径的一些下界。在得到这些结果的过程中，我们还纠正了Keevash和Ku（2006）在证明覆盖半径的下界时出现的一个错误。最后，我们讨论的多覆盖半径问题，在更一般的情况下，这提供了进一步研究的空间。
On completely regular self-dual codes with covering radius $ρ\leq 3$
该研究完成了覆盖半径ρ≤3的自对偶完全正则码的分类，明确了不同ρ下的码类构成、参数特征及交叉阵列，完善了相关编码理论结果。
The length function $\ell_q(r,R)$ is the smallest length of a $ q $-ary linear code with codimension (redundancy) $r$ and covering radius $R$. In this work, new upper bounds on $\ell_q(tR+1,R)$ are obtained in the following forms: \begin{equation*} \begin{split} &(a)~\ell_q(r,R)\le cq^{(r-R)/R}\cdot\sqrt[R]{\ln q},~ R\ge3,~r=tR+1,~t\ge1, &\phantom{(a)~} q\text{ is an arbitrary prime power},~c\text{ is independent of }q. \end{split} \end{equation*} \begin{equation*} \begin{split} &(b)~\ell_q(r,R)< 3.43Rq^{(r-R)/R}\cdot\sqrt[R]{\ln q},~ R\ge3,~r=tR+1,~t\ge1, &\phantom{(b)~} q\text{ is an arbitra…
Subspace coverings and generalized covering radii of generalized Zetterberg codes
本文研究广义Zetterberg码的广义覆盖半径，建立上界$2t+1$并给出下界，证明大$t$时达到最小值$t$，并确定二元码的第二、第三半径。
Covering-radius and Collinearity- Minimizing Pilots for Channel Estimation in TDD Systems
本文研究了在滑动窗口最新时隙恢复框架下，基于正交频分复用的时分双工系统中的试点设计，通过联合利用近期时隙中的时延-多普勒稀疏性，提出了一种几何感知的时间-频率联合试点分配方法，改进了覆盖半径和共线性控制性能。
On de Bruijn Covering Sequences and Arrays
本文研究 de Bruijn 覆盖序列与 dBCA，利用概率法给出面积上界，并通过折叠及循环码、自对偶序列等多种构造获得更短序列和更小面积阵列。
The length function $\ell_q(r,R)$ is the smallest length of a $q$-ary linear code of codimension (redundancy) $r$ and covering radius $R$. The $d$-length function $\ell_q(r,R,d)$ is the smallest length of a $q$-ary linear code with codimension $r$, covering radius $R$, and minimum distance $d$. By computer search in wide regions of $q$, we obtained following short codes of covering radius $R=3$: $[n,n-4,5]_q3$ quasi-perfect MDS codes, $[n,n-5,5]_q3$ quasi-perfect Almost MDS codes, and $[n,n-5,3]_q3$ codes. In computer search, we use the step-by-step leximatrix and inverse leximatrix algorithms…
Construction and Covering Properties of Constant-Dimension Codes
研究了用于随机网络编码中非相干纠错的恒维码。尽管提升Gabidulin码（简称KK码）得到的CDC是接近最优的，但给定最小距离的CDC的最大基数以及如何构造最优CDC都是一个悬而未决的问题。在本文中，我们首先构造了一类新的CDCs的基础上KK码，称为增广KK码，其基数大于以前提出的CDCs。然后，我们提出了一个低复杂度的解码算法，我们的增广KK码使用的KK码。我们的解码算法纠正更多的错误比有界子空间距离解码器利用我们的增广KK码的结构。在本文的其余部分，我们研究的覆盖性质的CDC。我们首先推导出一个覆盖半径给定的CDC的最小基数的界限，然后确定这个数量的渐近行为。此外，我们还证明了秩度量码的提升具有最大可能的覆盖半径，因此秩度量码的提升不是最优的填充CDC。最后，我们通过置换提升秩度量码来构造覆盖良好的CDC。
Symbol-pair codes were proposed for the application in high density storage systems, where it is not possible to read individual symbols. Yaakobi, Bruck and Siegel proved that the minimum pair-distance of binary linear cyclic codes satisfies $d_2 \geq \lceil 3d_H/2 \rceil$ and introduced $b$-symbol metric codes in 2016. In this paper covering codes in $b$-symbol metrics are considered. Some examples are given to show that the Delsarte bound and the Norse bound for covering codes in the Hamming metric are not true for covering codes in the pair metric. We give the redundancy bound on covering r…
The goal of the classic football-pool problem is to determine how many lottery tickets are to be bought in order to guarantee at least $n-r$ correct guesses out of a sequence of $n$ games played. We study a generalized (second-order) version of this problem, in which any of these $n$ games consists of two sub-games. The second-order version of the football-pool problem is formulated using the notion of generalized-covering radius, recently proposed as a fundamental property of linear codes. We consider an extension of this property to general (not necessarily linear) codes, and provide an asym…
New bounds for covering codes of radius 3 and codimension 3t+1
本文研究了q元线性码的长度函数，通过射影空间饱和集与提升构造，得到了半径为3且余维数为3t+1的覆盖码的新的构造性上界，且新界显著优于已知结果。
Quaternary Preparata codes are quasi-perfect covering codes in the Hamming metric
该论文证明四元Preparata码在汉明度量下对奇数长度参数为准完全覆盖码，覆盖半径为2、最小距离为4，并改进覆盖数上界，如$K_4(8,2)\le 256$。
Bounds on Covering Codes with the Rank Metric
本文研究了秩度量空间的几何性质和秩度量码的覆盖性质。首先建立了两个秩半径球相交的解析表达式，然后导出了多个秩半径球并的体积的上界。利用这些几何性质，我们推导出一个给定的秩覆盖半径的代码的最小基数的上限和下限。本文给出的秩度量码的几何性质和界对于秩度量码的设计、译码和性能分析具有重要意义。
For a class of graphs for which the Ramsey number $R(i,j)$ is upper bounded by $ci^aj^b$, for some constants $a,b,$ and $c$, it is shown that the clique covering scheme approximates the broadcast rate of every $n$-node index coding problem in the class within a multiplicative factor of $c^{\frac{1}{a+b+1}} n^{\frac{a+b}{a+b+1}}$ for every $n$. Using this theorem and some graph theoretic arguments, it is demonstrated that the broadcast rate of planar graphs, line graphs and fuzzy circular interval graphs is approximated by the clique covering scheme within a factor of $n^{\frac{2}{3}}$.
We study generalized covering radii, a fundamental property of linear codes that characterizes the trade-off between storage, latency, and access in linear data-query protocols such as PIR. We prove lower and upper bounds on the generalized covering radii of Reed-Muller codes, as well as finding their exact value in certain extreme cases. With the application to linear data-query protocols in mind, we also construct a covering algorithm that gets as input a set of points in space, and find a corresponding set of codewords from the Reed-Muller code that are jointly not farther away from the inp…
On linear $q$-ary completely regular codes with $ρ=2$ and dual antipodal
当对偶码对映时，刻画了所有覆盖半径为$ρ=2$的线性$q$ -ary完全正则码.这些完全正则码是覆盖半径为1的线性完全正则码的推广，它们都是可分类的。对于$ρ=2$，我们给出了我们已知的所有此类代码的列表。这也给出了两个重量线性对跖码的特征。
Let L be a Desarguesian 2-spread in the Grassmann graph $J_q(n,2)$. We prove that the collection of the 4-subspaces, which do not contain subspaces from L is a completely regular code in $J_q(n,4)$. Similarly, we construct a completely regular code in the Johnson graph $J(n,6)$ from the Steiner quadruple system of the extended Hamming code. We obtain several new completely regular codes covering radius 1 in the Grassmann graph $J_2(6,3)$ using binary linear programming.
Covering numbers of $L_p$-balls of convex sets and functions
证明了欧氏空间中凸函数类和凸集类覆盖数的界。以前的结果要求底层凸函数或凸集合是一致有界的。我们放松了这个假设，并用较弱的积分约束代替它。现有的结果可以恢复为我们的结果的特殊情况。
If the cyclic sequences of {face types} {at} all vertices in a map are the same, then the map is said to be a semi-equivelar map. In particular, a semi-equivelar map is equivelar if the faces are the same type. Homological quantum codes represent a subclass of topological quantum codes. In this article, we introduce {thirteen} new classes of quantum codes. These codes are associated with the following: (i) equivelar maps of type $ [k^k]$, (ii) equivelar maps on the double torus along with the covering of the maps, and (iii) semi-equivelar maps on the surface of \Echar{-1}, along with {their} c…
A covering code is a subset $\mathcal{C} \subseteq \{0,1\}^n$ with the property that any $z \in \{0,1\}^n$ is close to some $c \in \mathcal{C}$ in Hamming distance. For every $ε,δ>0$, we show a construction of a family of codes with relative covering radius $δ+ ε$ and rate $1 - \mathrm{H}(δ) $ with block length at most $\exp(O((1/ε) \log (1/ε)))$ for every $ε> 0$. This improves upon a folklore construction which only guaranteed codes of block length $\exp(1/ε^2)$. The main idea behind this proof is to find a distribution on codes with relatively small support such that most of these codes have…
A $λ$-fold $r$-packing (multiple radius-$r$ covering) in a Hamming metric space is a code $C$ such that the radius-$r$ balls centered in $C$ cover each vertex of the space by not more (not less, respectively) than $λ$ times. The well-known $r$-error-correcting codes correspond to the case $λ=1$, while in general multifold $r$-packing are related with list decodable codes. We (a) propose asymptotic bounds for the maximum size of a $q$-ary $2$-fold $1$-packing as $q$ grows; (b) prove that a $q$-ary distance-$2$ MDS code of length $n$ is an optimal $n$-fold $1$-packing if $q\ge 2n$; (c) derive an…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。