Constructing Linear Encoders with Good Spectra

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.

v5.5.5, no. 201408271350, 40 pages, 3 figures, extended version of the paper to be published in IEEE Transactions on Information Theory

References in corpus (8)

• Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels
• A Rank-Metric Approach to Error Control in Random Network Coding
• Threshold Saturation via Spatial Coupling: Why Convolutional LDPC Ensembles Perform so well over the BEC
• Weight Distributions of Regular Low-Density Parity-Check Codes over Finite Fields
• Linear-Codes-Based Lossless Joint Source-Channel Coding for Multiple-Access Channels
• Good Random Matrices over Finite Fields
• Constructing Linear Encoders with Good Spectra
• Second-Order Weight Distributions

Cited by in corpus (3)

• Weight Distributions of Regular Low-Density Parity-Check Codes over Finite Fields
• Good Random Matrices over Finite Fields
• Constructing Linear Encoders with Good Spectra