On Code Distances for a Class of Group Codes

The article considers group codes of group $G$, this being the direct product of second-order cyclic groups over field $K$, whose characteristic is different from 2. The code distances of the codes in question are investigated as a function of their dimension and of the number $n$. Assume that for code $I$ we have $KG=I\oplus\bar{I}$. On the basis of Berman?s hypothesis, for $\mathrm{dim}\bar{I}\leq q(n,k)$ (where $q(n,k)=\sum^k_{i=1} C_n^i$) the code distance of the code does not exceed $2^k$. It is shown in the paper that Berman's bound is exact for $n\leq 4$, but it becomes more and more crude as n increases. Explicit formulas are given for the numbers that refine this bound.