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Singleton-Optimal LRCs and Perfect LRCs via Cyclic and Constacyclic Codes 单最优lrc和基于循环码和恒循环码的完美lrc
Finite Fields Their Appl. Pub Date : 2023-03-11 DOI: 10.48550/arXiv.2303.06287
Weijun Fang, Fang-Wei Fu, Bin Chen, S. Xia
{"title":"Singleton-Optimal LRCs and Perfect LRCs via Cyclic and Constacyclic Codes","authors":"Weijun Fang, Fang-Wei Fu, Bin Chen, S. Xia","doi":"10.48550/arXiv.2303.06287","DOIUrl":"https://doi.org/10.48550/arXiv.2303.06287","url":null,"abstract":"Locally repairable codes (LRCs) have emerged as an important coding scheme in distributed storage systems (DSSs) with relatively low repair cost by accessing fewer non-failure nodes. Theoretical bounds and optimal constructions of LRCs have been widely investigated. Optimal LRCs via cyclic and constacyclic codes provide significant benefit of elegant algebraic structure and efficient encoding procedure. In this paper, we continue to consider the constructions of optimal LRCs via cyclic and constacyclic codes with long code length. Specifically, we first obtain two classes of $q$-ary cyclic Singleton-optimal $(n, k, d=6;r=2)$-LRCs with length $n=3(q+1)$ when $3 mid (q-1)$ and $q$ is even, and length $n=frac{3}{2}(q+1)$ when $3 mid (q-1)$ and $q equiv 1(bmod~4)$, respectively. To the best of our knowledge, this is the first construction of $q$-ary cyclic Singleton-optimal LRCs with length $n>q+1$ and minimum distance $d geq 5$. On the other hand, an LRC acheiving the Hamming-type bound is called a perfect LRC. By using cyclic and constacyclic codes, we construct two new families of $q$-ary perfect LRCs with length $n=frac{q^m-1}{q-1}$, minimum distance $d=5$ and locality $r=2$.","PeriodicalId":156673,"journal":{"name":"Finite Fields Their Appl.","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125427860","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
New advances in permutation decoding of first-order Reed-Muller codes 一阶Reed-Muller码的排列译码新进展
Finite Fields Their Appl. Pub Date : 2023-02-10 DOI: 10.48550/arXiv.2302.05189
J. J. Bernal, J. J. Sim'on
{"title":"New advances in permutation decoding of first-order Reed-Muller codes","authors":"J. J. Bernal, J. J. Sim'on","doi":"10.48550/arXiv.2302.05189","DOIUrl":"https://doi.org/10.48550/arXiv.2302.05189","url":null,"abstract":"In this paper we describe a variation of the classical permutation decoding algorithm that can be applied to any affine-invariant code with respect to certain type of information sets. In particular, we can apply it to the family of first-order Reed-Muller codes with respect to the information sets introduced in [2]. Using this algortihm we improve considerably the number of errors we can correct in comparison with the known results in this topic.","PeriodicalId":156673,"journal":{"name":"Finite Fields Their Appl.","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116044811","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Five infinite families of binary cyclic codes and their related codes with good parameters 五无限族二进制循环码及其相关的好参数码
Finite Fields Their Appl. Pub Date : 2023-01-16 DOI: 10.48550/arXiv.2301.06446
Hai Liu, Chengju Li, C. Ding
{"title":"Five infinite families of binary cyclic codes and their related codes with good parameters","authors":"Hai Liu, Chengju Li, C. Ding","doi":"10.48550/arXiv.2301.06446","DOIUrl":"https://doi.org/10.48550/arXiv.2301.06446","url":null,"abstract":"Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. Inspired by the recent work on binary cyclic codes published in IEEE Trans. Inf. Theory, vol. 68, no. 12, pp. 7842-7849, 2022, the objectives of this paper are the construction and analyses of five infinite families of binary cyclic codes with parameters $[n, k]$ and $(n-6)/3 leq k leq 2(n+6)/3$. Three of the five families of binary cyclic codes and their duals have a very good lower bound on their minimum distances and contain distance-optimal codes. The other two families of binary cyclic codes are composed of binary duadic codes with a square-root-like lower bound on their minimum distances. As a by-product, two infinite families of self-dual binary codes with a square-root-like lower bound on their minimum distances are obtained.","PeriodicalId":156673,"journal":{"name":"Finite Fields Their Appl.","volume":"156 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122025774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Constructions of cyclic codes and extended primitive cyclic codes with their applications 循环码和扩展原始循环码的构造及其应用
Finite Fields Their Appl. Pub Date : 2022-10-11 DOI: 10.48550/arXiv.2210.05170
Ziling Heng, Xinran Wang, Xiaoru Li
{"title":"Constructions of cyclic codes and extended primitive cyclic codes with their applications","authors":"Ziling Heng, Xinran Wang, Xiaoru Li","doi":"10.48550/arXiv.2210.05170","DOIUrl":"https://doi.org/10.48550/arXiv.2210.05170","url":null,"abstract":"Linear codes with a few weights have many nice applications including combinatorial design, distributed storage system, secret sharing schemes and so on. In this paper, we construct two families of linear codes with a few weights based on special polynomials over finite fields. The first family of linear codes are extended primitive cyclic codes which are affine-invariant. The second family of linear codes are reducible cyclic codes. The parameters of these codes and their duals are determined. As the first application, we prove that these two families of linear codes hold $t$-designs, where $t=2,3$. As the second application, the minimum localities of the codes are also determined and optimal locally recoverable codes are derived.","PeriodicalId":156673,"journal":{"name":"Finite Fields Their Appl.","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121444822","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Stabilizer quantum codes defined by trace-depending polynomials 由迹相关多项式定义的稳定器量子码
Finite Fields Their Appl. Pub Date : 2022-08-12 DOI: 10.48550/arXiv.2208.06187
C. Galindo, F. Hernando, Helena Mart'in-Cruz, D. Ruano
{"title":"Stabilizer quantum codes defined by trace-depending polynomials","authors":"C. Galindo, F. Hernando, Helena Mart'in-Cruz, D. Ruano","doi":"10.48550/arXiv.2208.06187","DOIUrl":"https://doi.org/10.48550/arXiv.2208.06187","url":null,"abstract":"Quantum error-correcting codes with good parameters can be constructed by evaluating polynomials at the roots of the polynomial trace. In this paper, we propose to evaluate polynomials at the roots of trace-depending polynomials (given by a constant plus the trace of a polynomial) and show that this procedure gives rise to stabilizer quantum error-correcting codes with a wider range of lengths than in other papers involving roots of the trace and with excellent parameters. Namely, we are able to provide new binary records and non-binary codes improving the ones available in the literature.","PeriodicalId":156673,"journal":{"name":"Finite Fields Their Appl.","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134329886","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Some 3-designs and shortened codes from binary cyclic codes with three zeros 一些由三个零的二进制循环码组成的3-设计和缩短码
Finite Fields Their Appl. Pub Date : 2022-06-30 DOI: 10.48550/arXiv.2206.15153
Can Xiang, Chunming Tang
{"title":"Some 3-designs and shortened codes from binary cyclic codes with three zeros","authors":"Can Xiang, Chunming Tang","doi":"10.48550/arXiv.2206.15153","DOIUrl":"https://doi.org/10.48550/arXiv.2206.15153","url":null,"abstract":"Linear codes and $t$-designs are interactive with each other. It is well known that some $t$-designs have been constructed by using certain linear codes in recent years. However, only a small number of infinite families of the extended codes of linear codes holding an infinite family of $t$-designs with $tgeq 3$ are reported in the literature. In this paper, we study the extended codes of the augmented codes of a class of binary cyclic codes with three zeros and their dual codes, and show that those codes hold $3$-designs. Furthermore, we obtain some shortened codes from the studied cyclic codes and explicitly determine their parameters. Some of those shortened codes are optimal or almost optimal.","PeriodicalId":156673,"journal":{"name":"Finite Fields Their Appl.","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121929584","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On non-monimial APcN permutations over finite fields of even characteristic 偶特征有限域上的非一元APcN排列
Finite Fields Their Appl. Pub Date : 2022-05-23 DOI: 10.48550/arXiv.2205.11418
Jaeseong Jeong, Namhun Koo, Soonhak Kwon
{"title":"On non-monimial APcN permutations over finite fields of even characteristic","authors":"Jaeseong Jeong, Namhun Koo, Soonhak Kwon","doi":"10.48550/arXiv.2205.11418","DOIUrl":"https://doi.org/10.48550/arXiv.2205.11418","url":null,"abstract":"Recently, a new concept called the $c$-differential uniformity was proposed by Ellingsen et al. (2020), which allows to simplify some types of differential cryptanalysis. Since then, finding functions having low $c$-differential uniformity has attracted the attention of many researchers. However it seems that, at this moment, there are not many non-monomial permutations having low $c$-differential uniformity. In this paper, we propose new classes of almost perfect $c$-nonlinear non-monomial permutations over a binary field.","PeriodicalId":156673,"journal":{"name":"Finite Fields Their Appl.","volume":"01 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124462580","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Constructions of near MDS codes which are optimal locally recoverable codes 近MDS码的构造是最优的局部可恢复码
Finite Fields Their Appl. Pub Date : 2022-04-24 DOI: 10.48550/arXiv.2204.11208
Xiaoru Li, Ziling Heng
{"title":"Constructions of near MDS codes which are optimal locally recoverable codes","authors":"Xiaoru Li, Ziling Heng","doi":"10.48550/arXiv.2204.11208","DOIUrl":"https://doi.org/10.48550/arXiv.2204.11208","url":null,"abstract":"A linear code with parameters $[n,k,n-k]$ is said to be almost maximum distance separable (AMDS for short). An AMDS code whose dual is also AMDS is referred to as an near maximum distance separable (NMDS for short) code. NMDS codes have nice applications in finite geometry, combinatorics, cryptography and data storage. In this paper, we first present several constructions of NMDS codes and determine their weight enumerators. In particular, some constructions produce NMDS codes with the same parameters but different weight enumerators. Then we determine the locality of the NMDS codes and obtain many families of distance-optimal and dimension-optimal locally repairable codes.","PeriodicalId":156673,"journal":{"name":"Finite Fields Their Appl.","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133448856","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 8
Hamming weight enumerators of multi-twisted codes with at most two non-zero constituents 最多有两个非零成分的多扭曲码的Hamming权枚举数
Finite Fields Their Appl. Pub Date : 2021-12-01 DOI: 10.1016/J.FFA.2021.101910
Varsha Chauhan, Anuradha Sharma
{"title":"Hamming weight enumerators of multi-twisted codes with at most two non-zero constituents","authors":"Varsha Chauhan, Anuradha Sharma","doi":"10.1016/J.FFA.2021.101910","DOIUrl":"https://doi.org/10.1016/J.FFA.2021.101910","url":null,"abstract":"","PeriodicalId":156673,"journal":{"name":"Finite Fields Their Appl.","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"118595008","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Application of Hermitian self-orthogonal GRS codes to some quantum MDS codes 厄米自正交GRS码在量子MDS码中的应用
Finite Fields Their Appl. Pub Date : 2021-12-01 DOI: 10.1016/J.FFA.2021.101901
Guanmin Guo, Ruihu Li, Yang Liu
{"title":"Application of Hermitian self-orthogonal GRS codes to some quantum MDS codes","authors":"Guanmin Guo, Ruihu Li, Yang Liu","doi":"10.1016/J.FFA.2021.101901","DOIUrl":"https://doi.org/10.1016/J.FFA.2021.101901","url":null,"abstract":"","PeriodicalId":156673,"journal":{"name":"Finite Fields Their Appl.","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"118248707","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 21
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