On Redundancy Reduction of Non-Recursive Second-Order Spectral-Null Codes

Abstract

The code design problem of non-recursive second-Order Spectral Null (2-OSN) codes is to convert balanced information words into 2-OSN words employing the minimum possible redundancy. Let $k$ be the balanced information word length. If $k\in \,\,2 {{I}}{{N}}$ then the 2-OSN coding scheme has length $n=k+r$ , with 2-OSN redundancy $r \in \,\,2 {{I}}{{N}}$ and $n\in \,\,4 {{I}}{{N}}$ . Here, we use a scheme with $r=2 \log k+\Theta (\log \log k)$ . The challenge is to reduce redundancy even further for any given $k$ . The idea is to exploit the degree of freedom to select from more than one possible 2-OSN encoding of a given balanced information word. To reduce redundancy, empirical results suggest that extra information $\delta _{k}=0.5 \log k+\Theta (\log \log k)$ is obtained. Thus, the proposed approach would give a smaller redundancy $r^{\prime }=1.5 \log k+\Theta (\log \log k)$ less than $r=2 \log k+\Theta (\log \log k)$ .