On adaptive linear programming decoding of nonbinary linear codes over prime fields

Abstract

In this work, we study linear programming (LP) decoding of nonbinary linear codes over prime fields. In particular, we develop a novel separation algorithm for valid inequalities describing the codeword polytope of the so-called constant-weight embedding of a single parity-check (SPC) code over any prime field. The algorithm has linear (in the length of the SPC code) complexity, is structurally different from the one for binary codes, and is based on the principle of dynamic programming. Furthermore, it is the basis of the proposed efficient (relaxed) adaptive LP (ALP) decoder for general (non-SPC) linear codes over any prime field, generalizing the well-known ALP decoding algorithm for binary codes. Numerical results show that the ALP decoding algorithm is very efficient compared to a static approach

Decoding Nonbinary LDPC Codes via Proximal-ADMM Approach (include convergence proofs)

“…Later, different cut-generating algorithms were designed in [26]- [28] to eliminate unexpected pseudo-codewords and improve the error-correction performance of LP decoding. Rosnes in [29] considered adaptive linear programming decoding of linear codes over prime fields and efficient separation of the underlying inequalities describing the decoding polytope which was done through dynamic programming. In addition, Liu et al in [30] and Bai et al in [10] proposed improved ADMM-based penalized decoding algorithms to enhance the error-correction performance of LP decoding in low SNR regions.…”

Efficient ADMM Decoder for Non-binary LDPC Codes with Codeword-Independent Performance

“…To resolve this problem, a coordinate ascent method was applied in [20] and [21] to solve the dual problem of the original LP problem [22] and [23]. In [24], Rosnes proposed adaptive linear programming decoding of linear codes over prime fields and dynamic programming was used in the separation algorithm of the decoding polytopes described by the resulting inequalities. Besides, Honda et al in [25] proposed a constant-weight embedding which used a binary vector to represent the element in F 2 q , however no further efficient algorithm was presented to tackle the resulting LP problem.…”