February  2007, 1(1): 111-130. doi: 10.3934/amc.2007.1.111

s-extremal additive $\mathbb F_4$ codes

1. 

Mathematics Department, School of Science and Engineering, Ateneo de Manila University, Loyola Heights, Quezon City, Philippines

2. 

XLIM, Université de Limoges, 123, Av. A. Thomas, 87000 Limoges, France

3. 

Department of Mathematics, University of Louisville, Louisvile, KY 40292

4. 

Department of Mathematics, 203 Avery Hall, University of Nebraska, Lincoln, NE 68588, United States

Received  June 2006 Revised  August 2006 Published  January 2007

Binary self-dual codes and additive self-dual codes over $\mathbb F_4$ have in common interesting properties, for example, Type I, Type II, shadows, etc. Recently Bachoc and Gaborit introduced the notion of $s$-extremality for binary self-dual codes, generalizing Elkies' study on the highest possible minimum weight of the shadows of binary self-dual codes. In this paper, we introduce a concept of $s$-extremality for additive self-dual codes over $\mathbb F_4$, give a bound on the length of these codes with even distance $d$, classify them up to minimum distance $d = 4$, give possible lengths and (shadow) weight enumerators for which there exist $s$-extremal codes with $5 \leq d \leq 11$ and give five $s$-extremal codes with $d = 7$. We construct four $s$-extremal codes of length $n = 13$ and minimum distance $d = 5$. We relate an $s$-extremal code of length $3d$ to another $s$-extremal code of that length, and produce extremal Type II codes from $s$-extremal codes.
Citation: Evangeline P. Bautista, Philippe Gaborit, Jon-Lark Kim, Judy L. Walker. s-extremal additive $\mathbb F_4$ codes. Advances in Mathematics of Communications, 2007, 1 (1) : 111-130. doi: 10.3934/amc.2007.1.111
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