# American Institute of Mathematical Sciences

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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