Daniele Bartoli, Matteo Bonini, Marco Timpanella
Minimal codes are a class of linear codes which gained interest in the last years, thanks to their connections to secret sharing schemes. In this paper we provide the weight distribution and the parameters of families of minimal codes recently introduced by C. Tang, Y. Qiu, Q. Liao, Z. Zhou, answering some open questions.
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
We provide new families of minimal codes in any characteristic. Also, an inductive construction of minimal codes is presented.
In this paper, we give a geometric characterization of minimal linear codes. In particular, we relate minimal linear codes to cutting blocking sets, introduced in a recent paper by Bonini and Borello. Using this characterization, we derive some bounds on the length and the distance of minimal codes, according to their dimension and the underlying field size. Furthermore, we show that the family of minimal codes is asymptotically good. Finally, we provide some geometrical constructions of minimal codes.
In this paper, we first study in detail the relationship between minimal linear codes and cutting blocking sets, which were recently introduced by Bonini and Borello, and then completely characterize minimal linear codes as cutting blocking sets. As a direct result, minimal projective codes of dimension $3$ and $t$-fold blocking sets with $t\ge 2$ in projective planes are identical objects. Some bounds on the parameters of minimal codes are derived from this characterization. This confirms a recent conjecture by Alfarano, Borello and Neri in [a geometric characterization of minimal codes and t…
Minimal codewords have applications in decoding linear codes and in cryptography. We study the maximum number of minimal codewords in binary linear codes of a given length and dimension. Improved lower and upper bounds on the maximum number are presented. We determine the exact values for the case of linear codes of dimension $k$ and length $k+2$ and for small values of the length and dimension. We also give a formula for the number of minimal codewords of linear codes of dimension $k$ and length $k+3$.
The main purpose of this paper is to further study the structure, parameters and constructions of the recently introduced minimal codes in the sum-rank metric. These objects form a bridge between the classical minimal codes in the Hamming metric, the subject of intense research over the past three decades partly because of their cryptographic properties, and the more recent rank-metric minimal codes. We prove some bounds on their parameters, existence results, and, via a tool that we name geometric dual, we manage to construct minimal codes with few weights. A generalization of the celebrated …
Singly-even self-dual codes with minimal shadow
本文研究了具有最小阴影的极值单偶自对偶码。对于特定的参数，我们证明不存在这样的代码。利用Rains \cite{Rains-asymptotic}的一个结果，证明了极值单偶自对偶码的长度是有界的.我们给出明确的界限的情况下，阴影是最小的。
In this paper we generalize constructions in two recent works of Ding, Heng, Zhou to any field $\mathbb{F}_q$, $q$ odd, providing infinite families of minimal codes for which the Ashikhmin-Barg bound does not hold.
The maximum number of minimal codewords in an $[n,k]-$code
本文导出了标题中的量的上、下界，并对$n$和$k.$的适度值进行了列表。
In this paper, we study a class of linear codes defined by characteristic functions of certain subsets of a finite field. We derive a sufficient and necessary condition for such a code to be a minimal linear code by a character-theoretical approach. We obtain new three-weight or four-weight minimal linear codes that do not satisfy the Ashikhmin-Barg condition by using partial difference sets. We show that our construction yields minimal linear codes that do not arise from cutting vectorial blocking sets, and also discuss their applications in secret sharing schemes.
We develop three approaches of combinatorial flavour to study the structure of minimal codes and cutting blocking sets in finite geometry, each of which has a particular application. The first approach uses techniques from algebraic combinatorics, describing the supports in a linear code via the Alon-Füredi Theorem and the Combinatorial Nullstellensatz. The second approach combines methods from coding theory and statistics to compare the mean and variance of the nonzero weights in a minimal code. Finally, the third approach regards minimal codes as cutting blocking sets and studies these using…
Minimal linear codes are in one-to-one correspondence with special types of blocking sets of projective spaces over a finite field, which are called strong or cutting blocking sets. In this paper we prove an upper bound on the minimal length of minimal codes of dimension $k$ over the $q$-element Galois field which is linear in both $q$ and $k$, hence improve the previous superlinear bounds. This result determines the minimal length up to a small constant factor. We also improve the lower and upper bounds on the size of so called higgledy-piggledy line sets in projective spaces and apply these …
Minimal linear codes are algebraic objects which gained interest in the last twenty years, due to their link with Massey's secret sharing schemes. In this context, Ashikhmin and Barg provided a useful and a quite easy to handle sufficient condition for a linear code to be minimal, which has been applied in the construction of many minimal linear codes. In this paper, we generalize some recent constructions of minimal linear codes which are not based on Ashikhmin-Barg's condition. More combinatorial and geometric methods are involved in our proofs. In particular, we present a family of codes ar…
Recently, some infinite families of minimal and optimal binary linear codes were constructed from simplicial complexes by Hyun {\em et al.} We extend this construction method to arbitrary posets. Especially, anti-chains are corresponded to simplicial complexes. In this paper, we present two constructions of binary linear codes from hierarchical posets of two levels. In particular, we determine the weight distributions of binary linear codes associated with hierarchical posets with two levels. Based on these results, we also obtain some optimal and minimal binary linear codes not satisfying the…
Recent advances on minimal codes
本文是一篇关于极小码最新进展的短篇综述，探讨了这一编码理论经典对象与其他数学领域（特别是有限射影几何）的联系，并指出这种联系如何激发了对极小码的重新关注，催生了新的构造与问题。
On diameter perfect constant-weight ternary codes
从陪集的二进制汉明码，我们构造直径完美的等重三元码的重量$n-1$（其中$n$是代码的长度）和距离3和5。距离5码的类具有未知的参数。关键词：等重码，三进制码，完美码，直径完美码，完美匹配，二进制码
Minimal additive codes and additive strong blocking sets
本文引入最小加法码的概念，通过建立与加法强阻塞集的一一对应关系，从几何角度研究其性质，并给出构造、存在性结果以及最小长度的上、下界和渐近界。
In recent years, many connections have been made between minimal codes, a classical object in coding theory, and other remarkable structures in finite geometry and combinatorics. One of the main problems related to minimal codes is to give lower and upper bounds on the length $m(k,q)$ of the shortest minimal codes of a given dimension $k$ over the finite field $\mathbb{F}_q$. It has been recently proved that $m(k, q) \geq (q+1)(k-1)$. In this note, we prove that $\liminf_{k \rightarrow \infty} \frac{m(k, q)}{k} \geq (q+ \varepsilon(q) )$, where $\varepsilon$ is an increasing function such that…
Linear error-correcting codes can be used for constructing secret sharing schemes; however finding in general the access structures of these secret sharing schemes and, in particular, determining efficient access structures is difficult. Here we investigate the properties of certain algebraic hypersurfaces over finite fields, whose intersection numbers with any hyperplane only takes a few values; these varieties give rise to $q$-divisible linear codes with at most $5$ weights. Furthermore, for $q$ odd these codes turn out to be minimal and we characterize the access structures of the secret sh…
In this article we present a class of codes with few weights arising from special type of linear sets. We explicitly show the weights of such codes, their weight enumerator and possible choices for their generator matrices. In particular, our construction yields also to linear codes with three weights and, in some cases, to almost MDS codes. The interest for these codes relies on their applications to authentication codes and secret schemes, and their connections with further objects such as association schemes and graphs.
This work investigates the structure of rank-metric codes in connection with concepts from finite geometry, most notably the $q$-analogues of projective systems and blocking sets. We also illustrate how to associate a classical Hamming-metric code to a rank-metric one, in such a way that various rank-metric properties naturally translate into the homonymous Hamming-metric notions under this correspondence. The most interesting applications of our results lie in the theory of minimal rank-metric codes, which we introduce and study from several angles. Our main contributions are bounds for the p…
We study sequences of linear or affine codes with uniform weight spectrum, i.e., a part of codewords with any fixed weight tends to zero. It is proved that a sequence of linear codes has a uniform weight spectrum if the number of vectors from codes with weight $1$ grows to infinity. We find an example of a sequence of linear codes such that the dimension of the code is the half of the codelength but it has not a uniform weight spectrum. This example generates eigenfunctions of the Fourier transform with minimal support and partial covering sets. Moreover, we generalize some MacWilliams-type id…
Linear complementary dual (LCD) codes are linear codes that intersect with their dual codes trivially. We study the largest minimum weight $d_2(n,k)$ among all binary LCD $[n,k]$ codes and the largest minimum weight $d_3(n,k)$ among all ternary LCD $[n,k]$ codes. The largest minimum weights $d_2(n,5)$ and $d_3(n,4)$ are partially determined. We also determine the largest minimum weights $d_2(n,n-5)$, $d_3(n,n-i)$ for $i \in \{2,3,4\}$, and $d_3(n,k)$ for $n \in \{11,12,\ldots,19\}$.
Strong blocking sets and minimal codes from expander graphs
本文结合常数度扩张图与渐近好码，在有限射影空间中显式构造了大小为 $O(qk)$ 的强阻塞集，从而首次给出了满足 $n=O(qk)$ 的极小线性码显式构造，解决了该领域的主要开放问题。
Counting the Number of Minimal Paths in Weighted Coloured-Edge Graphs
加权着色边图是一种图，其中每个边都被分配了一个正权重和一个离散颜色，并且可以用于对其中存在多种运输模式的运输和计算机网络进行建模。在这样的图中，路径通过它们在每种颜色中的总权重进行比较，从而产生从一个顶点到另一个顶点的最小路径的帕累托集合。本文给出了任意带权色边图的极小路集的基数的一个紧上界。此外，给出了加权双色边图中最小路的期望个数的一个界。
Minimal codewords have applications in decoding linear codes and in cryptography. We study the number of minimal codewords in binary linear codes that arise by appending a unit matrix to the adjacency matrix of a graph.
Linear Size Optimal q-ary Constant-Weight Codes and Constant-Composition Codes
一个最优的等重码$w$具有线性长度当且仅当它的距离$d$至少为$2w-1$。当$d\geq 2w$时，确定这种恒定组成或恒定重量代码的确切大小是微不足道的，但是$d=2w-1$的情况先前仅针对二进制和三进制恒定组成和恒定重量代码以及针对一些零星的实例而被解决。基于差三角集的一种新的推广，给出了权$w$和距离$2w-1$的拟循环最优常合成常重码的构造.结果，对于所有这样的长度足够大的码，确定了权重$w$和距离$2w-1$的最佳恒定组成码和最佳恒定权重码的大小。这解决了Etzion的一个开放问题。除了两种情况外，还确定了所有$w\leq 6$的重量$w$和距离$2w-1$的最佳常数组合码的大小。
Maximal quadrics over finite fields and minimal codewords of projective Reed-Muller codes
本文基于有限域上的最大二次曲面研究二阶投影Reed-Muller码的最小码字分类，证明了除F₂特例外，任何两个绝对不可约二次曲面若包含关系成立则必相等，并给出了最小码字的精确刻画及各重量下的数量。
We study the minimum number of minimal codewords in linear codes from the point of view of projective geometry. We derive bounds and in some cases determine the exact values. We also present an extension to minimal subcode supports.
A minimal code is a linear code where the only instance that a codeword has its support contained in the support of another codeword is when the codewords are scalar multiples of each other. Ashikhmin and Barg gave a sufficient condition for a code to be minimal, which led to much interest in constructing minimal codes that do not satisfy their condition. We consider a particular family of codes $\mathcal C_f$ when $f$ is the indicator function of a set of points, and prove a sufficient condition for $\mathcal C_f$ to be minimal and not satisfy Ashikhmin and Barg's condition based on certain g…
Minimal codes are being intensively studied in last years. $[n,k]_{q}$-minimal linear codes are in bijection with strong blocking sets of size $n$ in $PG(k-1,q)$ and a lower bound for the size of strong blocking sets is given by $(k-1)(q+1)\leq n$. In this note we show that all strong blocking sets of length 9 in $PG(3,2)$ are the hyperbolic quadrics $Q^{+}(3,2)$.
The set of minimal distances of the monoid of plus-minus weighted zero-sum sequences and applications to the characterization problem
本文延续正负加权零和序列幺半群的系统性研究，聚焦该类幺半群的重要算术不变量极小距离集合，并探讨其在刻画问题中的应用，研究受代数整数范数幺半群应用需求推动。
In the past few years, the study of receptive field codes has been of large interest to mathematicians. Here we give a complete characterization of receptive field codes realizable by connected receptive fields and we give the minimal embedding dimensions of these codes. In particular, we show that all connected codes are realizable in dimension at most 3. To our knowledge, this is the first family of receptive field codes for which the exact characterization and minimal embedding dimension is known.
The main result here is a characterisation of binary $2$-neighbour-transitive codes with minimum distance at least $5$ via their minimal subcodes, which are found to be generated by certain designs. The motivation for studying this class of codes comes primarily from their relationship to the class of completely regular codes. The results contained here yield many more examples of $2$-neighbour-transitive codes than previous classification results of families of $2$-neighbour-transitive codes. In the process, new lower bounds on the minimum distance of particular sub-families are produced. Sev…
Nonexistence of certain singly even self-dual codes with minimal shadow
已知对于$(n,d)=(24m+2,4m+4)$、$(24m+4,4m+4)$、$(24m+6,4m+4)$、$(24m+10,4m+4)$和$(24m+22,4m+6)$不存在具有最小阴影的极值单偶自对偶$[n,n/2,d]$码。本文研究了具有最小阴影的单偶自对偶码的最小权$d-2$。对于$n=24m+2$，$24m+4$和$24m+10$，我们证明了具有最小阴影的单偶自对偶$[n,n/2,4m+2]$码的权枚举数是唯一确定的，并且证明了对于$m \ge 155$，$m \ge 156$和$m \ge 160$，不存在具有最小阴影的单偶自对偶$[n,n/2,4m+2]$码.我们证明了具有最小阴影的单偶自对偶码的重量枚举数对于参数$[24m+6,12m+3,4m+2]$和$[24m+22,12m+11,4m+4]$不是唯一确定的。
In the 2017 paper by Dougherty, Kim, Ozkaya, Sok, and Solé about the linear programming bound for LCD codes the notion $\mathrm{LCD}[n,k]$ was defined for binary LCD $[n,k]$-codes. We find the formula for $\mathrm{LCD}[n,2]$.
Neural codes, represented as collections of binary strings called codewords, are used to encode neural activity. A code is called convex if its codewords are represented as an arrangement of convex open sets in Euclidean space. Previous work has focused on addressing the question: how can we tell when a neural code is convex? Giusti and Itskov identified a local obstruction and proved that convex neural codes have no local obstructions. The converse is true for codes on up to four neurons, but false in general. Nevertheless, we prove this converse holds for codes with up to three maximal codew…
New Inequalities for q-ary Constant-Weight Codes
利用双重计数方法证明了$q$ -元码的Delsarte不等式及其改进.将同样的方法应用于$q$元等重码，得到了$q$元等重码的新不等式.
Stanley-Reisner resolution of constant weight linear codes
给定一个等重线性码，我们研究了它的重量层次和Stanley-Reisner决议的相关拟阵视为单纯复形。我们还展示了更高的重量足以得出结论，代码是恒重的条件
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。