Disproof of a conjecture on the existence of balanced optimal covering codes

The minimum number of codewords in a binary code with length n and covering radius R is denoted by K(n,R), and corresponding codes are called optimal. A code with M words is said to be balanced in a given coordinate if the number of 0ʹs and 1ʹs in this coordinate are at least (...)M/2(...). A code is balanced if it is balanced in all coordinates. It has been conjectured that among optimal covering codes with given parameters there is at least one balanced code. By using a computational method for classifying covering codes, it is shown that there is no balanced code attaining K(9,1)=62.