Title :
Constructing Linear Encoders With Good Spectra
Author :
Shengtian Yang ; Honold, Thomas ; Yan Chen ; Zhaoyang Zhang ; Peiliang Qiu
Author_Institution :
Dept. of Inf. Sci. & Electron. Eng., Zhejiang Univ., Hangzhou, China
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: 1) lossless source coding; 2) channel coding; and 3) 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 (respectively, image) spectrum, it is proved that there exists a linear encoder not only with the same kernel (respectively, image) but also with a good joint spectrum. Thus, a good joint spectrum is the most important feature of a linear encoder.
Keywords :
combined source-channel coding; linear codes; matrix algebra; parity check codes; Gabidulin codes; average joint spectrum; code-spectrum approach; image spectra; inner encoder; input-output complete weight distribution; joint spectra; kernel spectra; linear encoders; lossless JSCC; lossless source coding; low-density generator matrix encoder; low-density parity-check code; maximum-rank-distance codes; optimal lossless joint source-channel coding; outer encoder; serial concatenation; Frequency modulation; Generators; Joints; Kernel; Linear codes; Code spectrum; Gabidulin codes; MacWilliams identities; linear codes; linear encoders; low-density generator matrix (LDGM); low-density parity-check (LDPC) codes; maximum-rank-distance (MRD) codes;
Journal_Title :
Information Theory, IEEE Transactions on
DOI :
10.1109/TIT.2014.2341560