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Adaptive Cut Generation Algorithm for Improved Linear Programming Decoding of Binary Linear Codes
Linear programming (LP) decoding approximates maximum-likelihood (ML)
decoding of a linear block code by relaxing the equivalent ML integer
programming (IP) problem into a more easily solved LP problem. The LP problem
is defined by a set of box constraints together with a set of linear
inequalities called "parity inequalities" that are derived from the constraints
represented by the rows of a parity-check matrix of the code and can be added
iteratively and adaptively. In this paper, we first derive a new necessary
condition and a new sufficient condition for a violated parity inequality
constraint, or "cut," at a point in the unit hypercube. Then, we propose a new
and effective algorithm to generate parity inequalities derived from certain
additional redundant parity check (RPC) constraints that can eliminate
pseudocodewords produced by the LP decoder, often significantly improving the
decoder error-rate performance. The cut-generating algorithm is based upon a
specific transformation of an initial parity-check matrix of the linear block
code. We also design two variations of the proposed decoder to make it more
efficient when it is combined with the new cut-generating algorithm. Simulation
results for several low-density parity-check (LDPC) codes demonstrate that the
proposed decoding algorithms significantly narrow the performance gap between
LP decoding and ML decoding
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