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Summary: Improved Lower Bounds for the Error Rate of
Linear Block Codes
Firouz Behnamfar, Fady Alajaji, and Tam´as Linder
Department Mathematics and Statistics
Department of Electrical and Computer Engineering
Queen's University
Kingston, Ontario Canada K7L 3N6ˇ firouz, fady, linder˘ @mast.queensu.ca
Abstract
We obtain two lower bounds on the error rate of linear binary block codes (under max-
imum likelihood decoding) over BPSK-modulated AWGN channels. We cast the problem
of finding a lower bound on the probability of a union as an optimization problem which
seeks to find the subset which maximizes a recent lower bound due to Kuai, Alajaji, and
Takahara that we will refer to as the KAT bound. Two variations of the KAT lower bound
are then derived. The first bound, the LB-f bound, requires the weight of the product of
the codewords with minimum weight in addition to their weight enumeration, while the
other bound, the LB-s bound (which is the main contribution of this paper), is algorithmic
and only needs the weight enumeration function of the code. The use of a subset of the
codebook to evaluate the KAT lower bound not only reduces computational complexity,
but also tightens this bound specially at low signal-to-noise (SNR) ratios. Numerical re-
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