Threshold Saturation for Nonbinary SC-LDPC Codes on the Binary Erasure Channel

• We analyze the asymptotic performance of nonbinary spatially coupled low-density parity-check (SC-LDPC) code ensembles defined over the general linear group on the binary erasure channel. In particular, we prove the threshold saturation of belief propagation decoding to the so-called potential threshold, using the proof technique based on potential functions introduced by Yedla et al., assuming that the potential function exists. We rewrite the density evolution of nonbinary SC-LDPC codes in an equivalent vector recursion form which is suited for the use of the potential function. We then discuss the existence of the potential function for the general case of vector recursions defined by multivariate polynomials, and give a method to construct it. We define a potential function in a slightly more general form than the one by Yedla et al., in order to make the technique based on potential functions applicable to the case of nonbinary LDPC codes. We show that the potential function exists if a solution to a carefully designed system of linear equations exists. Furthermore, we numerically show the existence of a solution to the system of linear equations for a large number of nonbinary LDPC code ensembles, which allows us to define their potential function and thus prove threshold saturation.

Ämnesord

TEKNIK OCH TEKNOLOGIER  -- Elektroteknik och elektronik -- Kommunikationssystem (hsv//swe)

ENGINEERING AND TECHNOLOGY  -- Electrical Engineering, Electronic Engineering, Information Engineering -- Communication Systems (hsv//eng)

TEKNIK OCH TEKNOLOGIER  -- Elektroteknik och elektronik (hsv//swe)

ENGINEERING AND TECHNOLOGY  -- Electrical Engineering, Electronic Engineering, Information Engineering (hsv//eng)

Nyckelord

binary erasure channel

nonbinary codes

threshold saturation

potential function

LDPC codes

spatial coupling

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