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LDPC codes associated with linear representations of geometries

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Abstract
We look at low density parity check codes over a finite field K associated with finite geometries T*(2) (K), where K is any subset of PG(2, q), with q = p(h), p not equal char K. This includes the geometry LU(3, q)(D), the generalized quadrangle T*(2)(K) with K a hyperoval, the affine space AG(3, q) and several partial and semi-partial geometries. In some cases the dimension and/or the code words of minimum weight are known. We prove an expression for the dimension and the minimum weight of the code. We classify the code words of minimum weight. We show that the code is generated completely by its words of minimum weight. We end with some practical considerations on the choice of K.
Keywords
linear representation, finite geometry, linear codes, LDPC codes, dimension, minimum distance

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Citation

Please use this url to cite or link to this publication:

Chicago
Vandendriessche, Peter. 2010. “LDPC Codes Associated with Linear Representations of Geometries.” Advances in Mathematics of Communications 4 (3): 405–417.
APA
Vandendriessche, P. (2010). LDPC codes associated with linear representations of geometries. ADVANCES IN MATHEMATICS OF COMMUNICATIONS, 4(3), 405–417.
Vancouver
1.
Vandendriessche P. LDPC codes associated with linear representations of geometries. ADVANCES IN MATHEMATICS OF COMMUNICATIONS. 2010;4(3):405–17.
MLA
Vandendriessche, Peter. “LDPC Codes Associated with Linear Representations of Geometries.” ADVANCES IN MATHEMATICS OF COMMUNICATIONS 4.3 (2010): 405–417. Print.
@article{1969420,
  abstract     = {We look at low density parity check codes over a finite field K associated with finite geometries T*(2) (K), where K is any subset of PG(2, q), with q = p(h), p not equal char K. This includes the geometry LU(3, q)(D), the generalized quadrangle T*(2)(K) with K a hyperoval, the affine space AG(3, q) and several partial and semi-partial geometries. In some cases the dimension and/or the code words of minimum weight are known. We prove an expression for the dimension and the minimum weight of the code. We classify the code words of minimum weight. We show that the code is generated completely by its words of minimum weight. We end with some practical considerations on the choice of K.},
  author       = {Vandendriessche, Peter},
  issn         = {1930-5346},
  journal      = {ADVANCES IN MATHEMATICS OF COMMUNICATIONS},
  keyword      = {linear representation,finite geometry,linear codes,LDPC codes,dimension,minimum distance},
  language     = {eng},
  number       = {3},
  pages        = {405--417},
  title        = {LDPC codes associated with linear representations of geometries},
  url          = {http://dx.doi.org/10.3934/amc.2010.4.405},
  volume       = {4},
  year         = {2010},
}

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