SISO Decoding of ℤ₄ Linear Kerdock and Preparata Codes

Abstract

Some nonlinear codes, such as Kerdock and Preparata codes, can be represented as binary images under the Gray map of linear codes over rings. This paper introduces MAP decoding of Kerdock and Preparata codes by working with their quaternary representation (linear codes over Z 4 ) with the complexity of O (𝑁 2 log 2 𝑁), where N is the code length in Z 4 . A sub-optimal bitwise APP decoder with good error-correcting performance and complexity of O (𝑁 log 2 𝑁) that is constructed using the decoder lifting technique is also introduced. This APP decoder extends upon the original lifting decoder by working with likelihoods instead of hard decisions and is not limited to Kerdock and Preparata code families. Simulations show that our novel decoders significantly outperform several popular decoders in terms of error rate.

A Survey on FEC Techniques for Industrial Wireless Communications

“…Linear block codes are less complex than non-linear ones and offer commonly good decoding performances. The non-linear block codes are characterized by a very high complexity that does not allow their consideration in low latency communications nowadays [40]. On the contrary, the respective counterpart shows many applicative examples in numerous standards.…”

“…Nevertheless, the recent binarization for Preparata and the Kerdock codes proposed in [40] opens new frontiers for their application within low latency communications.…”

“…The binarization of the Preparata and Kerdock codes combined with low complex decoding algorithms addresses future challenges for low latency communications [40]. In this first proposal, only the MAP decoding algorithm has been considered.…”

Decoding of Z2S Linear Generalized Kerdock Codes

“…Contribution. In our previous paper [24], we developed an SISO MAP decoding algorithm of linear Z 4 Kerdock codes with the complexity of O(N 2 log 2 N), where N is the code length of a quaternary code (i.e., the number of quaternary symbols in a codeword). Here, we extend this result to the family of generalized Kerdock codes.…”

“…Notation. In this paper, we employ the same notation as in [24]. We use bold letters to denote module elements, and by extension vectors, as every vector space is by definition a free module.…”