We investigate several types of linear codes constructed from two families $\tilde{\mathcal S}_q$ and $\tilde{\mathcal R}_q$ of maximal curves over finite fields recently constructed by Skabelund as cyclic covers of the Suzuki and Ree curves. Plane models for such curves are provided, and the Weierstrass semigroup $H(P)$ at an $\mathbb{F}_{q}$-rational point $P$ is shown to be symmetric.
We prove that the cyclic and constacyclic codes constructed by Grassl and Rötteler in arXiv:1502.05267 are generalised Reed-Solomon codes. This note can be considered as an addendum to that article. It can also be considered as an appendix to arXiv:2106.10180, where Conjecture 11 of arXiv:1502.0526, which was stated for Grassl-Rötteler codes, is proven for generalised Reed-Solomon codes. The content of this note, together with arXiv:2106.10180, therefore implies that Conjecture 11 from arXiv:1502.0526 is true.
Recently, Skabelund defined new maximal curves which are cyclic extensions of the Suzuki and Ree curves. Previously, the now well-known GK curves were found as cyclic extensions of the Hermitian curve. In this paper, we consider locally recoverable codes constructed from these new curves, complementing that done for the GK curve. Locally recoverable codes allow for the recovery of a single symbol by accessing only a few others which form what is known as a recovery set. If every symbol has at least two disjoint recovery sets, the code is said to have availability. Three constructions are descr…
Tsinghua-Berkeley Shenzhen Institute, Tsinghua Shenzhen International Graduate School（清华-伯克利深圳学院，清华大学深圳国际研究生院）
Neural decoders were introduced as a generalization of the classic Belief Propagation (BP) decoding algorithms, where the Trellis graph in the BP algorithm is viewed as a neural network, and the weights in the Trellis graph are optimized by training the neural network. In this work, we propose a novel neural decoder for cyclic codes by exploiting their cyclically invariant property. More precisely, we impose a shift invariant structure on the weights of our neural decoder so that any cyclic shift of inputs results in the same cyclic shift of outputs. Extensive simulations with BCH codes and pu…
Constacyclic codes over finite fields are a family of linear codes and contain cyclic codes as a subclass. Constacyclic codes are related to many areas of mathematics and outperform cyclic codes in several aspects. Hence, constacyclic codes are of theoretical importance. On the other hand, constacyclic codes are important in practice, as they have rich algebraic structures and may have efficient decoding algorithms. In this paper, two classes of constacyclic codes are constructed using a general construction of constacyclic codes with cyclic codes. The first class of constacyclic codes is moti…
我们利用有限域上的代数曲线和某些相关的Riemann-Roch空间构造线性网络码，并给出了获得其参数的方法。特别是，我们对待埃尔米特曲线和曲线与铃木和Ree组都具有最大数量的点的曲线，其各自的属。线性网络编码根据向量空间的基础来发送信息，并且该信息被接收为可能改变的向量空间的基础。作者声明：Frank R. Kschischang % \cite{DBLP:journals/tit/KoetterK08}介绍了一个度量的向量空间的集合，并表明，一个最小距离解码器，这个度量实现正确的解码，如果发送和接收的向量空间的交集的尺寸是足够大的。我们构造的向量空间在上述度量中具有从下到上的最小距离，这使得它们适合于线性网络编码。
The Schur product of evaluation codes and its application to CSS-T quantum codes and private information retrieval
IMUVA-Mathematics Research Institute, Universidad de Valladolid（瓦拉多利德大学 IMUVA 数学研究所）；Departamento de Matemáticas & Institut Universitari de Matemàtiques i Aplicacions de Castelló, Universitat Jaume I（卡斯特利翁大学数学系与数学及应用大学研究所）
In this Macaulay2 \cite{M2} package we define an object called {\it linear code}. We implement functions that compute basic parameters and objects associated with a linear code, such as generator and parity check matrices, the dual code, length, dimension, and minimum distance, among others. We define an object {\it evaluation code}, a construction which allows to study linear codes using tools of algebraic geometry and commutative algebra. We implement functions to generate important families of linear codes such as Hamming codes, cyclic codes, Reed--Solomon codes, Reed--Muller codes, Cartesi…
New Constructions of Binary Cyclic Codes with Both Relatively Large Minimum Distance and Dual Distance
State Key Laboratory of Cryptography and Digital Economy Security（密码学与数字经济安全国家重点实验室）；Key Laboratory of Cryptologic Technology and Information Security, Ministry of Education（教育部密码技术与信息安全重点实验室）；School of Cyber Science and Technology, Shandong University（山东大学网络空间科学院）；Quan Cheng Laboratory（泉城实验室）
The insertion-deletion codes were motivated to correct the synchronization errors. In this paper we prove several coordinate-ordering-free upper bounds on the insdel distances of linear codes, which are based on the generalized Hamming weights and the formation of minimum Hamming weight codewords. Our bounds are stronger than some previous known bounds. We apply these upper bounds to some cyclic codes and one algebraic-geometric code with any rearrangement of coordinate positions. Some strong upper bounds on the insdel distances of Reed-Muller codes with special coordinate-ordering are also gi…
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…
