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LDPC codes associated with linear representations of geometries. (English) Zbl 1218.51007
Low density parity check matrices, introduced by R. G. Gallager [IRE Trans. Inform. Theory IT-8, 21–28 (1962; Zbl 0107.11802)], are codes whose parity check matrices have a small percentage of nonzero elements. These codes have received a great deal of attention recently because they often outperform other codes and sometimes are very near the Shannon bound. The author examines LDPC codes over a field $$K$$ constructed from finite geometries $$T^*_2({\mathcal K})$$, where $${\mathcal K}$$ is any subset of $$PG(2,q)$$, $$q=p^h$$, $$p \neq \text{char}(K).$$ To define $$T^*_2({\mathcal K})$$ let $$\Pi$$ be a hyperplane in 3-dimensional projective space over $$F_q$$. In $$T^*_2({\mathcal K})$$, the points are the points of $$PG(3,q) \setminus PG(2,q)$$, the lines are the affine lines which are incident with a point of $${\mathcal K}$$ and the incidence relation is that of the ambient space. Examples include $$LU(3,q)^D$$, the generalized quadrangle $$T^*_2({\mathcal K})$$ with $${\mathcal K}$$ a hyperoval, the affine space $$AG(3,q)$$, and several partial and semi-partial geometries. The author gives a general setting for this study and provides a unified approach to previous results. In particular, he proves an expression for the dimension and minimum weight of the code, classifies code words of minimum weight, and proves that the code is generated by words of minimum weight.

##### MSC:
 51E20 Combinatorial structures in finite projective spaces 94B05 Linear codes, general 51E15 Finite affine and projective planes (geometric aspects)
LDPC
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