WCC 2005: Coding and Cryptography pp 14-21

# On Codes Correcting Symmetric Rank Errors

• Nina I. Pilipchuk
• Ernst M. Gabidulin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3969)

## Abstract

We study the capability of rank codes to correct so-called symmetric errors beyond the $$\left\lfloor \frac{d-1}{2}\right\rfloor$$ bound. If $$d\ge \frac{n+1}{2}$$, then a code can correct symmetric errors up to the maximal possible rank $$\lfloor\frac{n-1}{2}\rfloor$$. If $$d\le \frac{n}{2}$$, then the error capacity depends on relations between d and n. If $$(d+j)\nmid n,\;j=0,1,\dots,m-1$$, for some m, but (d+m) | n, then a code can correct symmetric errors up to rank $$\lfloor\frac{d+m-1}{2}\rfloor$$. In particular, one can choose codes correcting symmetric errors up to rank d–1, i.e., the error capacity for symmetric errors is about twice more than for general errors.

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