On Codes Correcting Symmetric Rank Errors

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.