Efficient Non-Recursive Design of Second-Order Spectral-Null Codes

Abstract

A new efficient design of second-order spectral-null (2-OSN) codes is presented. The new codes are obtained by applying the technique used to design parallel decoding balanced (i.e., 1-OSN) codes to the random walk method introduced by some of the authors for designing 2-OSN codes. This gives new non-recursive efficient code designs, which are less redundant than the code designs found in the literature. In particular, if $k \in {\mathrm{I}}\mskip -2.5mu{\mathrm{I}}\mskip -7mu{\mathrm{N}} $ is the length of a 1-OSN code then the new 2-OSN coding scheme has length $n=k+r \in {\mathrm{I}}\mskip -2.5mu{\mathrm{I}}\mskip -7mu{\mathrm{N}} $ with an extra redundancy of $r\simeq 2\log _{2}k+(1/2)\log _{2}\log _{2} k-0.174$ check bits, with $k$ and $r$ even and $n$ multiple of 4. The whole coding process requires $O(k\log k)$ bit operations and $O(k)$ bit memory elements.