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arXiv:cs/0702073

Tradeoff between decoding complexity and rate for codes on graphs

Pulkit Grover

Published Feb 13, 2007• 0 citations
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Abstract

We consider transmission over a general memoryless channel, with bounded decoding complexity per bit under message passing decoding. We show that the achievable rate is bounded below capacity if there is a finite success in the decoding in a specified number of operations per bit at the decoder for some codes on graphs. These codes include LDPC and LDGM codes. Good performance with low decoding complexity suggests strong local structures in the graphs of these codes, which are detrimental to the code rate asymptotically. The proof method leads to an interesting necessary condition on the code structures which could achieve capacity with bounded decoding complexity. We also show that if a code sequence achieves a rate epsilon close to the channel capacity, the decoding complexity scales at least as O(log(1/epsilon).

DOI
10.48550/arxiv.cs/0702073

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Khandekar and McEliece suggested the problem of per bit decoding complexity for capacity achieving sparse-graph codes as a function of their gap from the channel capacity. We consider the problem for the case of the binary symmetric channel. We derive a lower bound on this complexity for some codes on graphs for Belief Propagation decoding. For bounded degree LDPC and LDGM codes, any concatenation of the two, and punctured bounded-degree LDPC codes, this reduces to a lower bound of O (log (1/isin-)). The proof of this result leads to an interesting necessary condition on the code structures which could achieve capacity with bounded decoding complexity over BSC: the average edge-degree must converge to infinity while the average node-degree must be bounded. That is, one of the node degree distributions must have a finite mean and an infinite variance.
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An irregular LDGM-LDPC code is studied as a sub-code of an LDPC code with some randomly \emph{punctured} output-bits. It is shown that the LDGM-LDPC codes achieve rates arbitrarily close to the channel-capacity of the binary-input symmetric-output memoryless (BISOM) channel with bounded \emph{complexity}. The measure of complexity is the average-degree (per information-bit) of the check-nodes for the factor-graph of the code. A lower-bound on the average degree of the check-nodes of the irregular LDGM-LDPC codes is obtained. The bound does not depend on the decoder used at the receiver. The stability condition for decoding the irregular LDGM-LDPC codes over the binary-erasure channel (BEC) under iterative-decoding with message-passing is described.
An irregular LDGM-LDPC code is studied as a sub-code of an LDPC code with some randomly punctured output-bits. It is shown that the LDGM-LDPC codes achieve rates arbitrarily close to the channel-capacity of the binary-input symmetric-output memoryless (BISOM) channel with bounded complexity. The measure of complexity is the average-degree (per information-bit) of the check-nodes for the factor-graph of the code. A lower-bound on the average degree of the check-nodes of the irregular LDGM-LDPC codes is obtained. The bound does not depend on the decoder used at the receiver. The stability condition for decoding the irregular LDGM-LDPC codes over the binary-erasure channel (BEC) under iterative-decoding with message-passing is described.
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