Computer Science – Information Theory
Scientific paper
2009-09-17
Computer Science
Information Theory
29 pages, 2 figures, revised for the second round review of IEEE Transactions on Information Theory, Oct. 2010
Scientific paper
Linear codes with good joint spectra are good candidates for lossless joint source-channel coding (JSCC). However, in spite of their existence, it is still unknown how to construct them in practice. This paper is devoted to the construction of such codes. In particular, two families of linear codes are presented and proved to have good joint spectra. The first family is derived from Gabidulin codes, a class of maximum-rank-distance (MRD) codes. The second is constructed by a serial concatenation of an inner low-density generator matrix (LDGM) code and an outer low-density parity-check (LDPC) code. The general criteria of good linear codes are also reviewed and discussed for three cases: lossless source coding, channel coding, and lossless JSCC, respectively. In the framework of code-spectrum approach, it is shown that all these issues just correspond to the problems of constructing linear codes with good kernel spectra, good image spectra, and good joint spectra, respectively. Furthermore, it is shown that good joint spectra imply good kernel spectra and good image spectra. This fact thus establishes the fundamental role of linear codes with good joint spectra. Conversely, it is proved that, when the output (resp., input) alphabet is an elementary abelian group, there exist linear codes with not only good joint spectra but also the same kernel (resp., image) of any given linear code that has a good kernel (resp., image) spectrum.
Chen Yan
Honold Thomas
Qiu Pei-liang
Yang Shengtian
Zhang Zhaoyang
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