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1.
《Discrete Mathematics》2023,346(1):113126
New s-extremal extremal unimodular lattices in dimensions 38, 40, 42 and 44 are constructed from self-dual codes over by Construction A. In the process of constructing these codes, we obtain a self-dual code over . In addition, the code implies a code over . These codes have larger minimum weights than the previously known codes and codes, respectively. 相似文献
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3.
We study self-dual codes over the rings
and
. We define various weights and weight enumerators over these rings and describe the groups of invariants for each weight enumerator over the rings. We examine the torsion codes over these rings to describe the structure of self-dual codes. Finally we classify self-dual codes of small lengths over
. 相似文献
4.
The purpose of this paper is to improve the upper bounds of the minimum distances of self-dual codes over for lengths [22, 26, 28, 32–40]. In particular, we prove that there is no [22, 11, 9] self-dual code over , whose existence was left open in 1982. We also show that both the Hamming weight enumerator and the Lee weight enumerator
of a putative [24, 12, 10] self-dual code over are unique. Using the building-up construction, we show that there are exactly nine inequivalent optimal self-dual [18, 9,
7] codes over up to the monomial equivalence, and construct one new optimal self-dual [20, 10, 8] code over and at least 40 new inequivalent optimal self-dual [22, 11, 8] codes.
相似文献
5.
In this work, we focus on cyclic codes over the ring
\mathbbF2+u\mathbbF2+v\mathbbF2+uv\mathbbF2{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} , which is not a finite chain ring. We use ideas from group rings and works of AbuAlrub et.al. in (Des Codes Crypt 42:273–287,
2007) to characterize the ring
(\mathbbF2+u\mathbbF2+v\mathbbF2+uv\mathbbF2)/(xn-1){({{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2})/(x^n-1)} and cyclic codes of odd length. Some good binary codes are obtained as the images of cyclic codes over
\mathbbF2+u\mathbbF2+v\mathbbF2+uv\mathbbF2{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} under two Gray maps that are defined. We also characterize the binary images of cyclic codes over
\mathbbF2+u\mathbbF2+v\mathbbF2+uv\mathbbF2{{{\mathbb{F}}_2+u{\mathbb{F}}_2+v{\mathbb{F}}_2+uv{\mathbb{F}}_2}} in general. 相似文献
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We prove that every [n, k, d]
q
code with q ≥ 4, k ≥ 3, whose weights are congruent to 0, −1 or −2 modulo q and is extendable unless its diversity is for odd q, where .
相似文献
8.
Let Γ6 be the elliptic curve of degree 6 in PG(5, q) arising from a non-singular cubic curve of PG(2, q) via the canonical Veronese embedding
(1) If Γ6 (equivalently ) has n
GF(q)-rational points, then the associated near-MDS code has length n and dimension 6. In this paper, the case q = 5 is investigated. For q = 5, the maximum number of GF(q)-rational points of an elliptic curve is known to be equal to ten. We show that for an elliptic curve with ten GF(5)-rational points, the associated near-MDS code can be extended by adding two more points of PG(5, 5). In this way we obtain six non-isomorphic [12, 6]5 codes. The automorphism group of is also considered.
相似文献
9.
Let be a finite field of cardinality , which is a finite chain ring, and is an odd positive integer. For any , an explicit representation for the dual code of any -constacyclic code over of length is given. And some dual codes of -constacyclic codes over of length 14 are constructed. For the case of , all distinct self-dual -constacyclic codes over of length are determined. 相似文献
10.
In an earlier paper the authors studied simplex codes of type α and β over
and obtained some known binary linear and nonlinear codes as Gray images of these codes. In this correspondence, we study weight distributions of simplex codes of type α and β over
The generalized Gray map is then used to construct binary codes. The linear codes meet the Griesmer bound and a few non-linear codes are obtained that meet the Plotkin/Johnson bound. We also give the weight hierarchies of the first order Reed-Muller codes over
The above codes are also shown to satisfy the chain condition.A part of this paper is contained in his Ph.D. Thesis from IIT Kanpur, India 相似文献
11.
本文探索了环$R=Z_4[u]/\langle u2-2\rangle$ 上的几类斜多元循环码和多元循环码. 首先得到了环$R$上$(1,2u)$-多元循环码的生成多项式. 其次由定义的Gray映射得到了环$R$上$(1,2u)$- 多元循环码的Gray像是$Z_4$上的循环码或指数为2的逆循环码. 最后, 通过环$R$上$(1,2u)$- 多元循环码的一些例子来展示本文的主要结果. 相似文献
12.
Nuh Aydin Yasemin Cengellenmis Abdullah Dertli 《Designs, Codes and Cryptography》2018,86(6):1249-1255
In this paper, we study \(\lambda \)-constacyclic codes over the ring \(R=\mathbb {Z}_4+u\mathbb {Z}_4\) where \(u^{2}=1\), for \(\lambda =3+2u\) and \(2+3u\). Two new Gray maps from R to \(\mathbb {Z}_4^{3}\) are defined with the goal of obtaining new linear codes over \(\mathbb {Z}_4\). The Gray images of \(\lambda \)-constacyclic codes over R are determined. We then conducted a computer search and obtained many \(\lambda \)-constacyclic codes over R whose \(\mathbb {Z}_4\)-images have better parameters than currently best-known linear codes over \(\mathbb {Z}_4\). 相似文献
13.
With the help of some new results about weight enumerators of self-dual codes over
we investigate a class of double circulant codes over
, one of which leads to an extremal even unimodular 40–dimensional lattice. It is conjectured that there should be Nine more constructions of the Leech lattice 相似文献
14.
The Ramanujan Journal - In this paper, we extend to the function field setting the heuristics formerly developed by Conrey, Farmer, Keating, Rubinstein and Snaith, for the integral moments of... 相似文献
15.
Harold N. Ward 《Designs, Codes and Cryptography》2001,22(2):139-148
A divisibility theorem for codes meetingthe Griesmer bound is used to establish that there is no [207, 4, 165]code over GF(5). 相似文献