Constructing Linear Encoders with Good Spectra

TL;DR

This paper presents methods to construct linear encoders with optimal joint spectra, crucial for lossless joint source-channel coding, by leveraging Gabidulin codes and concatenated LDPC-based encoders.

Contribution

It introduces two new families of linear encoders with proven good joint spectra, advancing practical construction techniques for optimal JSCC.

Findings

Gabidulin code-based encoders have good joint spectra.

Concatenated LDPC and low-density generator matrix encoders are effective.

Good joint spectra ensure optimal performance in source and channel coding.

Abstract

Linear encoders with good joint spectra are suitable candidates for optimal lossless joint source-channel coding (JSCC), where the joint spectrum is a variant of the input-output complete weight distribution and is considered good if it is close to the average joint spectrum of all linear encoders (of the same coding rate). In spite of their existence, little is known on how to construct such encoders in practice. This paper is devoted to their construction. In particular, two families of linear encoders are presented and proved to have good joint spectra. The first family is derived from Gabidulin codes, a class of maximum-rank-distance codes. The second family is constructed using a serial concatenation of an encoder of a low-density parity-check code (as outer encoder) with a low-density generator matrix encoder (as inner encoder). In addition, criteria for good linear encoders are defined for three coding applications: lossless source coding, channel coding, and lossless JSCC. In the framework of the code-spectrum approach, these three scenarios correspond to the problems of constructing linear encoders with good kernel spectra, good image spectra, and good joint spectra, respectively. Good joint spectra imply both good kernel spectra and good image spectra, and for every linear encoder having a good kernel (resp., image) spectrum, it is proved that there exists a linear encoder not only with the same kernel (resp., image) but also with a good joint spectrum. Thus a good joint spectrum is the most important feature of a linear encoder.