Nonlinear Product Codes and Their Low Complexity Iterative Decoding

Abstract

This paper proposes encoding and decoding for nonlinear product codes and investigates the performance of nonlinear product codes. The proposed nonlinear product codes are constructed as N‐dimensional product codes where the constituent codes are nonlinear binary codes derived from the linear codes over higher order alphabets, for example, Preparata or Kerdock codes. The performance and the complexity of the proposed construction are evaluated using the well‐known nonlinear Nordstrom‐Robinson code, which is presented in the generalized array code format with a low complexity trellis. The proposed construction shows the additional coding gain, reduced error floor, and lower implementation complexity. The (64, 24, 12) nonlinear binary product code has an effective gain of about 2.5 dB and 1 dB gain at a BER of 10−6 when compared to the (64, 15, 16) linear product code and the (64, 24, 10) linear product code, respectively. The (256, 64, 36) nonlinear binary product code composed of two Nordstrom‐Robinson codes has an effective gain of about 0.7 dB at a BER of 10−5 when compared to the (256, 64, 25) linear product code composed of two (16, 8, 5) quasi‐cyclic codes.

Decoding of Z2S Linear Generalized Kerdock Codes

“…A detailed overview of different decoding algorithms for codes over rings is given in Section 2. Classical codes over rings are interesting as component codes in product and turbo coding schemes [16,22,23], but one of the main obstacles to their adoption in modern coding systems is the lack of efficient SISO decoding algorithms. As far as we are aware, generalized Kerdock codes do not yet have any soft decoders specifically designed to exploit their unique structure, so the main objective of this study is to address this gap by presenting maximum likelihood (ML) and maximum a posteriori (MAP) decoding algorithms for them.…”

SISO Decoding of ℤ4 Linear Kerdock and Preparata Codes

“…There is widespread use of the Nordstrom-Robinson code in digital systems [9,10]. A turbo product coding scheme with Nordstrom-Robinson as a component code was presented in [11,12]. Kerdock codes for limited feedback MIMO systems were presented in [13,14].…”