2 research outputs found
Sparsifying Parity-Check Matrices
Parity check matrices (PCMs) are used to define linear error correcting codes
and ensure reliable information transmission over noisy channels. The set of
codewords of such a code is the null space of this binary matrix.
We consider the problem of minimizing the number of one-entries in
parity-check matrices. In the maximum-likelihood (ML) decoding method, the
number of ones in PCMs is directly related to the time required to decode
messages. We propose a simple matrix row manipulation heuristic which alters
the PCM, but not the code itself. We apply simulated annealing and greedy local
searches to obtain PCMs with a small number of one entries quickly, i.e. in a
couple of minutes or hours when using mainstream hardware. The resulting
matrices provide faster ML decoding procedures, especially for large codes.Comment: This work was supported by Funda\c{c}\~ao para a Ci\^encia e
Tecnologia (FCT) ref. UID/CEC/50021/2019; European Union's Horizon 2020,
Marie Sk{\l}odowska-Curie Actions grant agreement No 690941; The DAAD-CRUP
Luso-German bilateral cooperation 2017-2018 research project MONO-EMC; The
DFG (project-ID: RU 1524/2-3). Jos{\'e} Rui Figueira acknowledges FCT grant
SFRH/BSAB/139892/201
Parity Check Matrices and Product Representations of Squares
Let NF(n, k, r) denote the maximum number of columns in an n-row matrix with entries ina finite field F in which each column has at most r nonzero entries and every k columns arelinearly independent over F. We obtain near-optimal upper bounds for NF(n, k, r) in the case k> r. Namely, we show that NF(n, k, r) # n r2 + cr k where c ij 43 for large k. Our method is based on a novel reduction of the problem to the extremal problem for cycles in graphs, and yields a fast algorithm for finding short linear dependences. We present additional applications of this method to problems in extremal hypergraph theory and combinatorial number theory