$m$ -ary Balanced Codes With Parallel Decoding.

An m -ary block code, m=2,3,4,\ldots , of length n \!\in \! \mathbfI\!\mathbfI\mskip -7mu\mathbfN is called balanced if, and only if, every codeword is balanced; that is, the real sum of the codeword components, or weight, is equal to \left \lfloor (m-1)n/2 \right \rfloor . This paper presents efficient encoding schemes to m -ary balanced codes with parallel (hence, fast) decoding. In fact, the decoding time complexity is O(1) digit operations. These schemes are a generalization to the m -ary alphabet of Knuth’s complementation method with parallel decoding. Let indicate the number of m -ary words of length n and weight . For any m \!\in \! {\mathbf{I}}\!{\mathbf{I}}\mskip -7mu{\mathbf{N}} , m\geq 2 , a simple implementation of the method is given which uses r \!\in \! \mathbfI\!\mathbfI\mskip -7mu\mathbfN check digits to balance k\leq \\binomr \left \lfloor (m-1)r/2 \right \rfloor \vphantom {RRm-\{m\bmod {2}+[(m-1)k]\bmod 2\}\}/(m-1) information digits with an encoding time complexity of O(mk\log mk) digit operations. A refined implementation of the parallel decoding method is also given with r check digits and k\leq (m^{r}-1)/(m-1) information digits, where the encoding time complexity is O(k\sqrt {\log _{m}k})$ . Thus, the proposed codes are less redundant than the $m$ -ary balanced codes with parallel decoding found in the literature and yet maintain the same complexity. [ABSTRACT FROM AUTHOR]