American Institute of Mathematical Sciences

Advanced
August  2007, 1(3): 357-398. doi: 10.3934/amc.2007.1.357

Additive self-dual codes over $\mathbb F_4$ with an automorphism of odd prime order

W. Cary Huffman 1,

1. 

Department of Mathematics and Statistics, Loyola University, Chicago, IL 60626, United States

Received  April 2007 Revised  June 2007 Published  July 2007

We present a general theory for decomposing additive self-dual codes over $\mathbbF_4$ that have an automorphism of odd prime order. We apply the decomposition to codes of length $n$ with $13\leq n\leq30$ and automorphisms of prime order $r$ with $5\leq r\leq23$. Using this decomposition we classify all extremal/optimal additive self-dual codes with certain parameters in this list. In the process, we find the first $(18$, 218, $7)$, $(24$, 224, $8)$, and $(28$, 228, $10)$ Type I codes. We also improve the lower bounds on the number of known extremal/optimal additive self-dual codes for some values of $n$ with $13\leq n\leq 30$.
Keywords: self-dual codes, automorphisms., Additive codes over $\bbF_4$.
Mathematics Subject Classification: 94B1.
Citation: W. Cary Huffman. Additive self-dual codes over $\mathbb F_4$ with an automorphism of odd prime order. Advances in Mathematics of Communications, 2007, 1 (3) : 357-398. doi: 10.3934/amc.2007.1.357
[1]

Padmapani Seneviratne, Martianus Frederic Ezerman. New quantum codes from metacirculant graphs via self-dual additive $\mathbb{F}_4$-codes. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2021073
[2]

Ken Saito. Self-dual additive $ \mathbb{F}_4 $-codes of lengths up to 40 represented by circulant graphs. Advances in Mathematics of Communications, 2019, 13 (2) : 213-220. doi: 10.3934/amc.2019014
[3]

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
[4]

Joaquim Borges, Steven T. Dougherty, Cristina Fernández-Córdoba. Characterization and constructions of self-dual codes over $\mathbb Z_2\times \mathbb Z_4$. Advances in Mathematics of Communications, 2012, 6 (3) : 287-303. doi: 10.3934/amc.2012.6.287
[5]

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
[6]

Gabriele Nebe, Wolfgang Willems. On self-dual MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 633-642. doi: 10.3934/amc.2016031
[7]

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
[8]

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
[9]

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
[10]

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
[11]

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
[12]

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
[13]

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
[14]

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
[15]

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
[16]

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
[17]

Somphong Jitman, San Ling, Ekkasit Sangwisut. On self-dual cyclic codes of length $p^a$ over $GR(p^2,s)$. Advances in Mathematics of Communications, 2016, 10 (2) : 255-273. doi: 10.3934/amc.2016004
[18]

Suat Karadeniz, Bahattin Yildiz. Double-circulant and bordered-double-circulant constructions for self-dual codes over $R_2$. Advances in Mathematics of Communications, 2012, 6 (2) : 193-202. doi: 10.3934/amc.2012.6.193
[19]

Steven T. Dougherty, Cristina Fernández-Córdoba, Roger Ten-Valls, Bahattin Yildiz. Quaternary group ring codes: Ranks, kernels and self-dual codes. Advances in Mathematics of Communications, 2020, 14 (2) : 319-332. doi: 10.3934/amc.2020023
[20]

Keita Ishizuka, Ken Saito. Construction for both self-dual codes and LCD codes. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2021070

2020 Impact Factor: 0.935

Tools

Metrics

  • PDF downloads (109)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy