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 ∈ 2IIN then the 2-OSN coding scheme has length n = k + r, with 2-OSN redundancy r ∈ 2IIN and n ∈ 4IIN. Here, we use a scheme with r = 2 log k + Θ(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 δk = 0.5 log k + Θ(log log k) is obtained. Thus, the proposed approach would give a smaller redundancy r0 = 1.5 log k + Θ(log log k) less than r = 2log k + Θ(log log k).

Keywords

• Balanced codes
• high order spectral null codes
• Knuth’s complementation method
• parallel decoding scheme
• optical and magnetic recording