$m$ -ary Balanced Codes With Parallel Decoding


An $m$ -ary block code, $m=2,3,4,ldots $ , of length $n !in ! {mathbf{I}}!{mathbf{I}}mskip -7mu{mathbf{N}} $ 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 $binom{n}{ w}_{m}$ indicate the number of $m$ -ary words of length $n$ and weight $w !in !{0,1,ldots ,(m-1)n}$ . For any $m !in ! {mathbf{I}}!{mathbf{I}}mskip -7mu{mathbf{N}} $ , $mgeq 2$ , a simple implementation of the method is given which uses $r !in ! {mathbf{I}}!{mathbf{I}}mskip -7mu{mathbf{N}} $ check digits to balance $kleq {binom{{r}}{ { left lfloor{ (m-1)r/2 }right rfloor }}_{vphantom {R_{R_{}}}m}-{mbmod {2}+[(m-1)k]bmod 2}}/(m-1)$ information digits with an encoding time complexity of $O(mklog _{m}k)$ digit operations. A refined implementation of the parallel decoding method is also given with $r$ check digits and $kleq (m^{r}-1)/(m-1)$ information digits, where the encoding time complexity is $O(ksqrt {log _{m}k})$ . 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.

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