Infinite families of $t$-designs from a type of five-weight codes

New infinite families of p-ary weakly regular bent functions

The characterization and construction of bent functions are challenging problems. The paper generalizes the constructions of Boolean bent functions by Mesnager \cite{M2014}, Xu et al. \cite{XCX2015} and $p$-ary bent functions by Xu et al. \cite{XC2015} to the construction of $p$-ary weakly regular bent functions and presents new infinite families of $p$-ary weakly regular bent functions from some known weakly regular bent functions (square functions, Kasami functions, and the Maiorana-McFarland class of bent functions). Further, new infinite families of $p$-ary bent idempotents are obtained.

Infinite families of $2$-designs from a class of non-binary Kasami cyclic codes

Combinatorial $t$-designs have been an important research subject for many years, as they have wide applications in coding theory, cryptography, communications and statistics. The interplay between coding theory and $t$-designs has been attracted a lot of attention for both directions. It is well known that a linear code over any finite field can be derived from the incidence matrix of a $t$-design, meanwhile, that the supports of all codewords with a fixed weight in a code also may hold a $t$-design. In this paper, by determining the weight distribution of a class of linear codes derived from…

关于球填充界的最优距离二元线性码的无穷多族

Infinitely many families of distance-optimal binary linear codes with respect to the sphere packing bound

本文解决了编码理论中75年未决的开放问题：证明了存在关于球填充界、具有任意大最小距离的最优距离线性码的无穷族，并给出了若干副产品族。

On new infinite families of completely regular and completely transitive codes

In two previous papers we constructed new families of completely regular codes by concatenation methods. Here we determine cases in which the new codes are completely transitive. For these cases we also find the automorphism groups of such codes. For the remaining cases, we show that the codes are not completely transitive assuming an upper bound on the order of the monomial automorphism groups, according to computational results.

Infinite families of $2$-designs from a class of linear codes related to Dembowski-Ostrom functions

Due to their important applications to coding theory, cryptography, communications and statistics, combinatorial $t$-designs have been attracted lots of research interest for decades. The interplay between coding theory and $t$-designs has on going for many years. As we all known, $t$-designs can be used to derive linear codes over any finite field, as well as the supports of all codewords with a fixed weight in a code also may hold a $t$-design. In this paper, we first construct a class of linear codes from cyclic codes related to Dembowski-Ostrom functions. By using exponential sums, we then…

与$O^{-}_{}(2n,q)$相关的生成平方变元Kloosterman和幂矩的无穷族递归公式

Infinite Families of Recursive Formulas Generating Power Moments of Ternary Kloosterman Sums with Square Arguments Associated with $O^{-}_{}(2n,q)$

本文构造了八个关于特殊正交群$SO^{-}(2n,q)$的极大抛物子群的具有双陪集的三元线性码的无限族。这里，${q}$是3的幂。在此基础上，我们得到了四个关于带平方变元的Kloosterman和的幂次矩的无穷族递归公式和四个关于其偶次幂次矩的无穷族递归公式.这是通过Pless幂矩恒等式和利用与正交群$O^{-}(2n,q)$的$"$高斯和的求值相关的双陪集上的指数和的显式来完成的。

不存在好的无限族toric码

There are no good infinite families of toric codes

本研究通过证明更一般的Szemerédi型结果，证实Soprunov与Soprunova提出的好的无限族toric码并不存在，明确了高维格点稠密子集的超立方体结构性质。

Infinite families of linear codes supporting more $t$-designs

Tang and Ding [IEEE IT 67 (2021) 244-254] studied the class of narrow-sense BCH codes $\mathcal{C}_{(q,q+1,4,1)}$ and their dual codes with $q=2^m$ and established that the codewords of the minimum (or the second minimum) weight in these codes support infinite families of 4-designs or 3-designs. Motivated by this, we further investigate the codewords of the next adjacent weight in such codes and discover more infinite classes of $t$-designs with $t=3,4$. In particular, we prove that the codewords of weight $7$ in $\mathcal{C}_{(q,q+1,4,1)}$ support $4$-designs when $m \geqslant 5$ is odd and $…

无限家族的图与任意矩阵的稳定完成，第一部分

Infinite families of graphs and stable completion of arbitrary matrices, Part I

本文提出了一种确定性构造方法，通过平方和层次结构实现任意秩矩阵在无限家族图上的稳定完成。

生成Kloosterman和幂矩的递归公式的无限族：O^-（2n，2^r）情形

Infinite families of recursive formulas generating power moments of Kloosterman sums: O^- (2n, 2^r) case

本文构造了八个关于特殊正交群$SO^-(2n,2^r)$的极大抛物子群的二元线性码的双陪集无限族。然后我们得到了四个无限族的Kloosterman和的幂矩的递归公式，以及四个二维Kloosterman和的幂矩的递归公式。这是通过Pless幂矩恒等式和利用与正交群$O^-(2n,2^r)$的“高斯和”的求值有关的双陪集上的指数和的显式来完成的

生成Kloosterman和幂矩的无穷族递归公式：O\^{+}（2n，2 r）情形

Infinite Families of Recursive Formulas Generating Power Moments of Kloosterman Sums: O\^{+}(2n, 2r) Case

本文构造了关于正交群O^+（2n，2 ^r）的某个极大抛物子群的四个具有双陪集的二元线性码的无限族.这里q是2的幂。然后得到了Kloosterman和的幂矩和二维Kloosterman和的幂矩关于码中权的频率的两个无穷族递推公式。这是通过Pless幂矩恒等式和利用与正交群O^+（2n，2 ^r）的“高斯和”的求值相关的双陪集上的指数和的显式来实现的。

Infinite Families of Quantum-Classical Hybrid Codes

Hybrid codes simultaneously encode both quantum and classical information into physical qubits. We give several general results about hybrid codes, most notably that the quantum codes comprising a genuine hybrid code must be impure and that hybrid codes can always detect more errors than comparable quantum codes. We also introduce the weight enumerators for general hybrid codes, which we then use to derive linear programming bounds. Finally, inspired by the construction of some families of nonadditive codes, we construct several infinite families of genuine hybrid codes with minimum distance t…

Infinite families of near MDS codes holding $t$-designs

An $[n, k, n-k+1]$ linear code is called an MDS code. An $[n, k, n-k]$ linear code is said to be almost maximum distance separable (almost MDS or AMDS for short). A code is said to be near maximum distance separable (near MDS or NMDS for short) if the code and its dual code both are almost maximum distance separable. The first near MDS code was the $[11, 6, 5]$ ternary Golay code discovered in 1949 by Golay. This ternary code holds $4$-designs, and its extended code holds a Steiner system $S(5, 6, 12)$ with the largest strength known. In the past 70 years, sporadic near MDS codes holding $t$-d…

生成Kloosterman和幂矩的无穷族递归公式：辛情形

Infinite Families of Recursive Formulas Generating Power Moments of Kloosterman Sums: Symplectic Case

本文构造了关于辛群Sp（2 n，q）的某个极大抛物子群的两个具有双陪集的二元线性码无限族，其中q是2的幂.然后得到了Kloosterman和的幂矩以及二维Kloosterman和的幂矩关于码中权的频率的无穷族递推公式。这是通过Pless幂矩恒等式和利用与辛群Sp（2n，q）的“高斯和”的计算有关的双陪集上的指数和的显式来完成的。

Three New Infinite Families of Optimal Locally Repairable Codes from Matrix-Product Codes

Locally repairable codes have become a key instrument in large-scale distributed storage systems. This paper focuses on the construction of locally repairable codes with $(r,δ)$-locality that achieve the equality in the Singleton-type bound. We use matrix-product codes to propose two infinite families of $q$-ary optimal $(r,δ)$ locally repairable codes of lengths up to $q^2+q$. The ingredients in the matrix-product codes are either linear maximum distance separable codes or optimal locally repairable codes of small lengths. Further analysis and refinement yield a construction of another infini…

Infinite families of 3-designs from APN functions

Combinatorial $t$-designs have nice applications in coding theory, finite geometries and several engineering areas. The objective of this paper is to study how to obtain $3$-designs with $2$-transitive permutation groups. The incidence structure formed by the orbits of a base block under the action of the general affine groups, which are $2$-transitive, is considered. A characterization of such incidence structure to be a $3$-design is presented, and a sufficient condition for the stabilizer of a base block to be trivial is given. With these general results, infinite families of $3$-designs ar…

两类BCH码的对偶码

The dual codes of two families of BCH codes

本文通过研究两类BCH码的对偶码参数，构造了无限族MDS码和几乎MDS码，解决了2022年提出的猜想，并证明了相关码及其对偶码承载无限族3-设计，同时提供了最佳参数的二元、三元和四元码。

生成迹非零平方变元的三值Kloosterman和幂矩的无穷族递归公式：$O(2n+1,2^{r})$情形

Infinite Families of Recursive Formulas Generating Power Moments of Ternary Kloosterman Sums with Trace Nonzero Square Arguments: $O(2n+1,2^{r})$ Case

本文构造了$O(2n+1,q)$中关于特殊正交群$SO(2n+1,q)$的某个极大抛物子群的四个具有双陪集的三元线性码的无限族.这里$q$是三的幂。然后得到了两个无穷族递归公式，一个产生具有$``$迹非零平方参数的Kloosterman和的幂矩，另一个产生它们的偶次幂矩.这两个族都用与$O(2n+1,q)$中的那些双陪集相关联的码和与辛群$Sp(2n,q)$中的类似双陪集相关联的码中的权重的频率来表示。这是通过Pless幂矩恒等式和利用与正交群$O(2n+1,q)$的$"$高斯和的求值相关的双陪集上的指数和的显式来完成的。

一类优良自正交准循环码无限族的构造

A Construction of Infinite Families of Good Self-Orthogonal Quasi-Cyclic Codes

本文通过域扩张上的组成码构造了无限族准循环码，使其在欧几里得和内积下自正交，并计算了维数和最小距离的平方根型下界，进而利用CSS构造得到参数良好的量子纠错码。

利用单纯复形构造环上的无穷多族最优码与极小码

Infinite families of optimal and minimal codes over rings using simplicial complexes

该研究利用单纯复形在非单位非交换环扩环上构造无穷多族码，表征其参数、确定李重量分布，进一步得到大量满足极小性或距离最优的有限域线性码。

关于二次APN函数无限族的Fourier谱

On the Fourier Spectra of the Infinite Families of Quadratic APN Functions

定义在奇数次有限域上的二次函数几乎弯曲（AB）当且仅当它几乎完全非线性（APN）。对于偶数度的情况，函数的傅立叶谱中的值与APN属性之间没有明显的关系。在这篇文章中，我们计算的傅里叶谱的新的四项家庭的APN功能。有了这个结果，所有已知的APN函数的无限族现在都有了它们的傅里叶谱，因此计算了它们的非线性。

无限族常循环码支持3设计及其在编码理论中的应用

Infinite families of constacyclic codes supporting 3-designs and their applications in coding theory

本文研究了支持无限族3设计的λ-常循环码，推导了其参数和重量分布，并应用于构造纠缠辅助量子误差校正码和局部可恢复码。

几类新的具有任意维数且支撑t-设计的NMDS码的无限族

Several new infinite families of NMDS codes with arbitrary dimensions supporting $t$-designs

本文构造了多类具有任意维数且支撑t-设计的NMDS码无限族，确定其重量分布，推广了固定维数结果并解决了Heng-Wang猜想。

New Infinite Families of Perfect Quaternion Sequences and Williamson Sequences

We present new constructions for perfect and odd perfect sequences over the quaternion group $Q_8$. In particular, we show for the first time that perfect and odd perfect quaternion sequences exist in all lengths $2^t$ for $t\geq0$. In doing so we disprove the quaternionic form of Mow's conjecture that the longest perfect $Q_8$-sequence that can be constructed from an orthogonal array construction is of length 64. Furthermore, we use a connection to combinatorial design theory to prove the existence of a new infinite class of Williamson sequences, showing that Williamson sequences of length $2…

具有类平方根下界的四族无限三元循环码

Four infinite families of ternary cyclic codes with a square-root-like lower bound

受已有二元循环码研究启发，本文构造并分析了四族长度为$n=3^m-1$（$m$为奇数）的无限三元循环码及其对偶码，二者最小距离均具类平方根下界，且包含距离最优循环码。

源自Ding猜想的六族二元码

Six Families of Binary Codes Arising from Ding's Conjectures

本文研究Ding关于布尔函数二元线性码的六个未解猜想，通过构造无限族、证明权重限制及给出反例，完全解决了其中两个猜想。