Pub Date : 2018-05-28DOI: 10.13069/jacodesmath.427968
Kazuki Kumegawa, T. Maruta
We prove the non--existence of $[g_q(4,d),4,d]_q$ codes for $d=2q^3-rq^2-2q+1$ for $3 le r le (q+1)/2$, $q ge 5$; $d=2q^3-3q^2-3q+1$ for $q ge 9$; $d=2q^3-4q^2-3q+1$ for $q ge 9$; and $d=q^3-q^2-rq-2$ with $r=4, 5$ or $6$ for $q ge 9$, where $g_q(4,d)=sum_{i=0}^{3} leftlceil d/q^i rightrceil$. This yields that $n_q(4,d) = g_q(4,d)+1$ for $2q^3-3q^2-3q+1 le d le 2q^3-3q^2$, $2q^3-5q^2-2q+1 le d le 2q^3-5q^2$ and $q^3-q^2-rq-2 le d le q^3-q^2-rq$ with $4 le r le 6$ for $q ge 9$ and that $n_q(4,d) ge g_q(4,d)+1$ for $2q^3-rq^2-2q+1 le d le 2q^3-rq^2-q$ for $3 le r le (q+1)/2$, $q ge 5$ and $2q^3-4q^2-3q+1 le d le 2q^3-4q^2-2q$ for $q ge 9$, where $n_q(4,d)$ denotes the minimum length $n$ for which an $[n,4,d]_q$ code exists.
我们证明的不存在 $[g_q(4,d),4,d]_q$ 代码 $d=2q^3-rq^2-2q+1$ 为了 $3 le r le (q+1)/2$， $q ge 5$； $d=2q^3-3q^2-3q+1$ 为了 $q ge 9$； $d=2q^3-4q^2-3q+1$ 为了 $q ge 9$;和 $d=q^3-q^2-rq-2$ 有 $r=4, 5$ 或 $6$ 为了 $q ge 9$，其中 $g_q(4,d)=sum_{i=0}^{3} leftlceil d/q^i rightrceil$。结果是 $n_q(4,d) = g_q(4,d)+1$ 为了 $2q^3-3q^2-3q+1 le d le 2q^3-3q^2$， $2q^3-5q^2-2q+1 le d le 2q^3-5q^2$ 和 $q^3-q^2-rq-2 le d le q^3-q^2-rq$ 有 $4 le r le 6$ 为了 $q ge 9$ 这就是 $n_q(4,d) ge g_q(4,d)+1$ 为了 $2q^3-rq^2-2q+1 le d le 2q^3-rq^2-q$ 为了 $3 le r le (q+1)/2$， $q ge 5$ 和 $2q^3-4q^2-3q+1 le d le 2q^3-4q^2-2q$ 为了 $q ge 9$，其中 $n_q(4,d)$ 表示最小长度。 $n$ 为了什么? $[n,4,d]_q$ 代码存在。
{"title":"Non-existence of some 4-dimensional Griesmer codes over finite fields","authors":"Kazuki Kumegawa, T. Maruta","doi":"10.13069/jacodesmath.427968","DOIUrl":"https://doi.org/10.13069/jacodesmath.427968","url":null,"abstract":"We prove the non--existence of $[g_q(4,d),4,d]_q$ codes for $d=2q^3-rq^2-2q+1$ for $3 le r le (q+1)/2$, $q ge 5$; $d=2q^3-3q^2-3q+1$ for $q ge 9$; $d=2q^3-4q^2-3q+1$ for $q ge 9$; and $d=q^3-q^2-rq-2$ with $r=4, 5$ or $6$ for $q ge 9$, where $g_q(4,d)=sum_{i=0}^{3} leftlceil d/q^i rightrceil$. This yields that $n_q(4,d) = g_q(4,d)+1$ for $2q^3-3q^2-3q+1 le d le 2q^3-3q^2$, $2q^3-5q^2-2q+1 le d le 2q^3-5q^2$ and $q^3-q^2-rq-2 le d le q^3-q^2-rq$ with $4 le r le 6$ for $q ge 9$ and that $n_q(4,d) ge g_q(4,d)+1$ for $2q^3-rq^2-2q+1 le d le 2q^3-rq^2-q$ for $3 le r le (q+1)/2$, $q ge 5$ and $2q^3-4q^2-3q+1 le d le 2q^3-4q^2-2q$ for $q ge 9$, where $n_q(4,d)$ denotes the minimum length $n$ for which an $[n,4,d]_q$ code exists.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47232360","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}
Pub Date : 2018-03-25DOI: 10.13069/jacodesmath.671815
A. Melakhessou, N. Aydin, K. Guenda
In this paper, we study skew constacyclic codes over the ring $mathbb{Z}_{q}R$ where $R=mathbb{Z}_{q}+umathbb{Z}_{q}$, $q=p^{s}$ for a prime $p$ and $u^{2}=0$. We give the definition of these codes as subsets of the ring $mathbb{Z}_{q}^{alpha}R^{beta}$. Some structural properties of the skew polynomial ring $ R[x,theta]$ are discussed, where $ theta$ is an automorphism of $R$. We describe the generator polynomials of skew constacyclic codes over $ R $ and $mathbb{Z}_{q}R$. Using Gray images of skew constacyclic codes over $mathbb{Z}_{q}R$ we obtained some new linear codes over $mathbb{Z}_4$. Further, we have generalized these codes to double skew constacyclic codes over $mathbb{Z}_{q}R$.
本文研究了环$mathbb{Z}_{q}R$上的偏常环码，其中$R=mathbb{Z}_{q}+umathbb{Z}_{q}$, $q=p^{s}$为素数$p$和$u^{2}=0$。我们给出了这些码作为环$mathbb{Z}_{q}^{alpha}R^{beta}$子集的定义。讨论了歪多项式环$ R[x,theta]$的一些结构性质，其中$ theta$是$R$的自同构。我们描述了$ R $和$mathbb{Z}_{q}R$上的偏常环码的生成器多项式。利用$mathbb{Z}_{q}R$上偏常环码的灰度图像，得到了$mathbb{Z}_4$上新的线性码。进一步，我们将这些码推广到$mathbb{Z}_{q}R$上的双斜常环码。
{"title":"$mathbb{Z}_{q}(mathbb{Z}_{q}+umathbb{Z}_{q})-$ linear skew constacyclic codes","authors":"A. Melakhessou, N. Aydin, K. Guenda","doi":"10.13069/jacodesmath.671815","DOIUrl":"https://doi.org/10.13069/jacodesmath.671815","url":null,"abstract":"In this paper, we study skew constacyclic codes over the ring $mathbb{Z}_{q}R$ where $R=mathbb{Z}_{q}+umathbb{Z}_{q}$, $q=p^{s}$ for a prime $p$ and $u^{2}=0$. We give the definition of these codes as subsets of the ring $mathbb{Z}_{q}^{alpha}R^{beta}$. Some structural properties of the skew polynomial ring $ R[x,theta]$ are discussed, where $ theta$ is an automorphism of $R$. We describe the generator polynomials of skew constacyclic codes over $ R $ and $mathbb{Z}_{q}R$. Using Gray images of skew constacyclic codes over $mathbb{Z}_{q}R$ we obtained some new linear codes over $mathbb{Z}_4$. Further, we have generalized these codes to double skew constacyclic codes over $mathbb{Z}_{q}R$.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47575885","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}
Pub Date : 2016-02-07DOI: 10.13069/jacodesmath.09585
Somphong Jitman, P. Udomkavanich
The class of 1-generator quasi-abelian codes over finite fields is revisited. Alternative and explicit characterization and enumeration of such codes are given. An algorithm to find all 1-generator quasi-abelian codes is provided. Two 1-generator quasi-abelian codes whose minimum distances are improved from Grassl’s online table are presented.
{"title":"One–generator quasi–abelian codes revisited","authors":"Somphong Jitman, P. Udomkavanich","doi":"10.13069/jacodesmath.09585","DOIUrl":"https://doi.org/10.13069/jacodesmath.09585","url":null,"abstract":"The class of 1-generator quasi-abelian codes over finite fields is revisited. Alternative and explicit characterization and enumeration of such codes are given. An algorithm to find all 1-generator quasi-abelian codes is provided. Two 1-generator quasi-abelian codes whose minimum distances are improved from Grassl’s online table are presented.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2016-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66233241","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}
Pub Date : 2014-06-15DOI: 10.13069/JACODESMATH.560406
Somphong Jitman, S. Ling
Self-dual and complementary dual cyclic/abelian codes over finite fields form important classes of linear codes that have been extensively studied due to their rich algebraic structures and wide applications. In this paper, abelian codes over Galois rings are studied in terms of the ideals in the group ring ${ GR}(p^r,s)[G]$, where $G$ is a finite abelian group and ${ GR}(p^r,s)$ is a Galois ring. Characterizations of self-dual abelian codes have been given together with necessary and sufficient conditions for the existence of a self-dual abelian code in ${ GR}(p^r,s)[G]$. A general formula for the number of such self-dual codes is established. In the case where $gcd(|G|,p)=1$, the number of self-dual abelian codes in ${ GR}(p^r,s)[G]$ is completely and explicitly determined. Applying known results on cyclic codes of length $p^a$ over ${ GR}(p^2,s)$, an explicit formula for the number of self-dual abelian codes in ${ GR}(p^2,s)[G]$ are given, where the Sylow $p$-subgroup of $G$ is cyclic. Subsequently, the characterization and enumeration of complementary dual abelian codes in ${ GR}(p^r,s)[G]$ are established. The analogous results for self-dual and complementary dual cyclic codes over Galois rings are therefore obtained as corollaries.
{"title":"Self-dual and complementary dual abelian codes over Galois rings","authors":"Somphong Jitman, S. Ling","doi":"10.13069/JACODESMATH.560406","DOIUrl":"https://doi.org/10.13069/JACODESMATH.560406","url":null,"abstract":"Self-dual and complementary dual cyclic/abelian codes over finite fields form important classes of linear codes that have been extensively studied due to their rich algebraic structures and wide applications. In this paper, abelian codes over Galois rings are studied in terms of the ideals in the group ring ${ GR}(p^r,s)[G]$, where $G$ is a finite abelian group and ${ GR}(p^r,s)$ is a Galois ring. Characterizations of self-dual abelian codes have been given together with necessary and sufficient conditions for the existence of a self-dual abelian code in ${ GR}(p^r,s)[G]$. A general formula for the number of such self-dual codes is established. In the case where $gcd(|G|,p)=1$, the number of self-dual abelian codes in ${ GR}(p^r,s)[G]$ is completely and explicitly determined. Applying known results on cyclic codes of length $p^a$ over ${ GR}(p^2,s)$, an explicit formula for the number of self-dual abelian codes in ${ GR}(p^2,s)[G]$ are given, where the Sylow $p$-subgroup of $G$ is cyclic. Subsequently, the characterization and enumeration of complementary dual abelian codes in ${ GR}(p^r,s)[G]$ are established. The analogous results for self-dual and complementary dual cyclic codes over Galois rings are therefore obtained as corollaries.","PeriodicalId":37029,"journal":{"name":"Journal of Algebra Combinatorics Discrete Structures and Applications","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2014-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66232853","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}
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