# \$m\$ -ary Balanced Codes With Parallel Decoding

PROJECT TITLE :

\$m\$ -ary Balanced Codes With Parallel Decoding

ABSTRACT:

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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