Polarity-balanced codes

Balanced bipolar codes consist of sequences in which the symbols '-1' and '+1' appear equally often. Several generalizations to larger alphabets have been considered in literature. For example, for the q-ary alphabet {-q + 1, -q + 3, ..., q - 1}, known concepts are symbol balancing, i.e., all alphabet symbols appear equally often in each codeword, and charge balancing, i.e., the symbol sum in each codeword equals zero. These notions are equivalent for the bipolar case, but not for q > 2. In this paper, a third perspective is introduced, called polarity balancing, where the number of positive symbols equals the number of negative symbols in each codeword. The minimum redundancy of such codes is determined and a generalization of Knuth's celebrated bipolar balancing algorithm is proposed.