Abstract
We look at low density parity check codes over a finite field $\mathbb K$ associated with finite geometries $T$2*$(\mathcal K)$, where $\mathcal K$ is any subset of PG$(2,q)$, with $q=p$h, $p$≠char$\mathbb K$. This includes the geometry $LU(3,q)$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. 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 $\mathcal K$.
Highlights
Introduced by Gallager [5], low density parity check (LDPC) codes are used frequently today due to their excellent empirical performance under belief-propagation/sum-product decoding [19]
In this article we study the general problem of codes associated with linear representations of geometries when char K = p
When char K = p, we compute the minimum weight, the rank and the code rate of the code, we classify the code words of minimum weight and we prove that every code word is a linear combination of the code words of minimum weight
Summary
Introduced by Gallager [5], low density parity check (LDPC) codes are used frequently today due to their excellent empirical performance under belief-propagation/sum-product decoding [19]. We denote by C the code generated by all plane words. Let ∈ L be a line in the plane at infinity, containing exactly k points of K.
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