Construction of new self-dual codes over GF(5) using skew-Hadamard matrices

Previous Article
A review of the available construction methods for Golomb rulers
AMC Home
This Issue
Next Article
A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$

August 2009, 3(3): 251-263. doi: 10.3934/amc.2009.3.251

Construction of new self-dual codes over $GF(5)$ using skew-Hadamard matrices

Christos Koukouvinos ^1, and Dimitris E. Simos ^1,

1.	Department of Mathematics, National Technical University of Athens, Zografou 15773, Athens, Greece, Greece

Received February 2009 Revised May 2009 Published August 2009

Abstract
Related Papers

In this paper, we give optimal self-dual codes over $GF(5)$ for lengths $24$, $40$, $48$ and $56$. In particular, new inequivalent $[48, 24]$ and $[56, 28]$ self-dual codes over $GF(5)$ whose minimum weights are $14$ and $16$, are constructed using skew-Hadamard matrices of order $24$ and $28$, thus improving the only known quadratic double circulant self-dual codes of length $48$ and $56$. Moreover, $[80, 40]$ and $[88, 44]$ self-dual codes whose minimum weights are $17$ and $19$ over $GF(5)$, are constructed for the first time. These codes are derived from skew-Hadamard matrices of order $40$ and $44$, respectively. Finally, a new $[56, 28, 17]$ self-dual code is constructed over $GF(7)$ having the highest minimum weight among $[56, 28]$ self-dual codes. This new optimal code is constructed from a skew-Hadamard-matrix of order $28$, for the first time.

Keywords: Self-dual codes, construction..

Mathematics Subject Classification: Primary: 94B05, 05B20; Secondary: 94B2.

Citation: Christos Koukouvinos, Dimitris E. Simos. Construction of new self-dual codes over $GF(5)$ using skew-Hadamard matrices. Advances in Mathematics of Communications, 2009, 3 (3) : 251-263. doi: 10.3934/amc.2009.3.251

[1]	Hyun Jin Kim, Heisook Lee, June Bok Lee, Yoonjin Lee. Construction of self-dual codes with an automorphism of order $p$. Advances in Mathematics of Communications, 2011, 5 (1) : 23-36. doi: 10.3934/amc.2011.5.23
[2]	Gabriele Nebe, Wolfgang Willems. On self-dual MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 633-642. doi: 10.3934/amc.2016031
[3]	Delphine Boucher. Construction and number of self-dual skew codes over $\mathbb{F}_{p^2}$. Advances in Mathematics of Communications, 2016, 10 (4) : 765-795. doi: 10.3934/amc.2016040
[4]	Masaaki Harada, Akihiro Munemasa. Classification of self-dual codes of length 36. Advances in Mathematics of Communications, 2012, 6 (2) : 229-235. doi: 10.3934/amc.2012.6.229
[5]	Stefka Bouyuklieva, Anton Malevich, Wolfgang Willems. On the performance of binary extremal self-dual codes. Advances in Mathematics of Communications, 2011, 5 (2) : 267-274. doi: 10.3934/amc.2011.5.267
[6]	Nikolay Yankov, Damyan Anev, Müberra Gürel. Self-dual codes with an automorphism of order 13. Advances in Mathematics of Communications, 2017, 11 (3) : 635-645. doi: 10.3934/amc.2017047
[7]	Masaaki Harada, Akihiro Munemasa. On the covering radii of extremal doubly even self-dual codes. Advances in Mathematics of Communications, 2007, 1 (2) : 251-256. doi: 10.3934/amc.2007.1.251
[8]	Stefka Bouyuklieva, Iliya Bouyukliev. Classification of the extremal formally self-dual even codes of length 30. Advances in Mathematics of Communications, 2010, 4 (3) : 433-439. doi: 10.3934/amc.2010.4.433
[9]	Bram van Asch, Frans Martens. Lee weight enumerators of self-dual codes and theta functions. Advances in Mathematics of Communications, 2008, 2 (4) : 393-402. doi: 10.3934/amc.2008.2.393
[10]	Bram van Asch, Frans Martens. A note on the minimum Lee distance of certain self-dual modular codes. Advances in Mathematics of Communications, 2012, 6 (1) : 65-68. doi: 10.3934/amc.2012.6.65
[11]	Masaaki Harada, Katsushi Waki. New extremal formally self-dual even codes of length 30. Advances in Mathematics of Communications, 2009, 3 (4) : 311-316. doi: 10.3934/amc.2009.3.311
[12]	Katherine Morrison. An enumeration of the equivalence classes of self-dual matrix codes. Advances in Mathematics of Communications, 2015, 9 (4) : 415-436. doi: 10.3934/amc.2015.9.415
[13]	Minjia Shi, Daitao Huang, Lin Sok, Patrick Solé. Double circulant self-dual and LCD codes over Galois rings. Advances in Mathematics of Communications, 2019, 13 (1) : 171-183. doi: 10.3934/amc.2019011
[14]	Suat Karadeniz, Bahattin Yildiz. New extremal binary self-dual codes of length $68$ from $R_2$-lifts of binary self-dual codes. Advances in Mathematics of Communications, 2013, 7 (2) : 219-229. doi: 10.3934/amc.2013.7.219
[15]	Amita Sahni, Poonam Trama Sehgal. Enumeration of self-dual and self-orthogonal negacyclic codes over finite fields. Advances in Mathematics of Communications, 2015, 9 (4) : 437-447. doi: 10.3934/amc.2015.9.437
[16]	Steven T. Dougherty, Cristina Fernández-Córdoba. Codes over $\mathbb{Z}_{2^k}$, Gray map and self-dual codes. Advances in Mathematics of Communications, 2011, 5 (4) : 571-588. doi: 10.3934/amc.2011.5.571
[17]	Masaaki Harada. Note on the residue codes of self-dual $\mathbb{Z}_4$-codes having large minimum Lee weights. Advances in Mathematics of Communications, 2016, 10 (4) : 695-706. doi: 10.3934/amc.2016035
[18]	Ekkasit Sangwisut, Somphong Jitman, Patanee Udomkavanich. Constacyclic and quasi-twisted Hermitian self-dual codes over finite fields. Advances in Mathematics of Communications, 2017, 11 (3) : 595-613. doi: 10.3934/amc.2017045
[19]	Lars Eirik Danielsen. Graph-based classification of self-dual additive codes over finite fields. Advances in Mathematics of Communications, 2009, 3 (4) : 329-348. doi: 10.3934/amc.2009.3.329
[20]	Ilias S. Kotsireas, Christos Koukouvinos, Dimitris E. Simos. MDS and near-MDS self-dual codes over large prime fields. Advances in Mathematics of Communications, 2009, 3 (4) : 349-361. doi: 10.3934/amc.2009.3.349

2018 Impact Factor: 0.879

American Institute of Mathematical Sciences