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conference paper

Phase Transitions for Mutual Information

Kumar, Krishna
•
Kumar, Raj
•
Pakzad, Payam
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2010
Proc. 6th Intl. Symposium on Turbo Codes and Iterative Information Processing (ISTC - 2010)
6th Intlernational Symposium on Turbo Codes and Iterative Information Processing (ISTC)

We consider ensembles of binary linear error correcting codes, obtained by sampling each column of the generator matrix G or parity check matrix H independently from the set of all binary vectors of weight d (of appropriate dimension). We investigate the circumstances under which the mutual information between a randomly chosen codeword and the vector obtained after its transmission over a binary input memoryless symmetric channel (BIMSC) C is exactly n times the capacity of C, where n is the length of the code. For several channels such as the binary symmetric channel (BSC) and the binary-input additive white Gaussian noise (AWGN) channel, we prove that the probability of this event has a threshold behaviour, depending on whether n/k is smaller than a certain quantity (that depends on the particular channel C and d), where k is the number of source bits. To show this, we prove a generalization of the following well-known theorem: the expectation of the size of the right kernel of G has a phase transition from 1 to infinity, depending on whether or not n/k is smaller than a certain quantity depending on the chosen ensemble.

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Phase Transition for Mutual Information.pdf

Type

Postprint

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http://purl.org/coar/version/c_ab4af688f83e57aa

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openaccess

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318.83 KB

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Adobe PDF

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