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Shortened Array Codes of Large Girth
One approach to designing structured low-density parity-check (LDPC) codes
with large girth is to shorten codes with small girth in such a manner that the
deleted columns of the parity-check matrix contain all the variables involved
in short cycles. This approach is especially effective if the parity-check
matrix of a code is a matrix composed of blocks of circulant permutation
matrices, as is the case for the class of codes known as array codes. We show
how to shorten array codes by deleting certain columns of their parity-check
matrices so as to increase their girth. The shortening approach is based on the
observation that for array codes, and in fact for a slightly more general class
of LDPC codes, the cycles in the corresponding Tanner graph are governed by
certain homogeneous linear equations with integer coefficients. Consequently,
we can selectively eliminate cycles from an array code by only retaining those
columns from the parity-check matrix of the original code that are indexed by
integer sequences that do not contain solutions to the equations governing
those cycles. We provide Ramsey-theoretic estimates for the maximum number of
columns that can be retained from the original parity-check matrix with the
property that the sequence of their indices avoid solutions to various types of
cycle-governing equations. This translates to estimates of the rate penalty
incurred in shortening a code to eliminate cycles. Simulation results show that
for the codes considered, shortening them to increase the girth can lead to
significant gains in signal-to-noise ratio in the case of communication over an
additive white Gaussian noise channel.Comment: 16 pages; 8 figures; to appear in IEEE Transactions on Information
Theory, Aug 200
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