A Family Of Low Density Matrices In Lagrangian–Grassmannian

Abstract

The aim of this paper is twofold. First, we show a connection between the Lagrangian- Grassmannian variety geometry defined over a finite field with q elements and the q-ary Low-Density Parity- Check codes. Second, considering the Lagrangian-Grassmannian variety as a linear section of the Grassmannian variety, we prove that there is a unique linear homogeneous polynomials family, up to linear combination, such that annuls the set of its rational points.

Iterative methods to build LG-matrices and Applications

“…where Z(•) ⊆ P E denotes the set of zeros of the given polynomials, α ∈ I(n − 1, m), β ∈ I(n + 1, m), α st ∈ I(n − 2, 2n). Moreover the set of rational points is defined by L(n, E)(F q ) = Z Q α ,β , Π αst , X q α − X α α∈I(n,2n) In [4], it is shown that if h ∈ E * suct that h(L(n, E)(F q )) = 0, then h ∈ Π αrs |α rs ∈ I(n − 2, 2n) .…”

“…Is important we notice that throughout this paper, when α belongs a set of indexes I(s, m), it is assumed that belongs up a permutation that orders it properly. We consider a symplectic vector space and a subspace, a way to establish if this subspace is isotropic, is by means of a matrix and the Plücker coordinates, such matrix is given in [4]. In this paper for n ≥ 4 arbitrary positive integer, m = 2k − 2, 2 ≤ k ≤ r and r = n+2 2 partition is given in the whole I(n−2, 2n) with sets T m called triangles and for each triangle T m defines a configuration and the incidence matrix of this configuration corresponds to a matrix L k , we give a series of rules to form a matrix M = L k • • • L 2 which turns out to be a "canonical form" of the Lagrangian-Grassmannian variety L(n, E) where E is a simplectic vector space such that dim E = 2n.…”

“…An important fact is that they have the properties of persymmetric and similarity. (2) We introduce a family of Low Density Parity Check codes (LDPC-codes), these are a generalization of the codes considered in [4]. For our new family of codes we propose an optimal decoding algorithm using "Efficient encoders based on approximate lower triangulations" developed by Richardson-Urbanke see [7].…”

“…, h } ⊂ ( n E) * are its rows. As H annuls the rational points of L(n, E)(F q ), by the Theorem 12 of [4] we have h 1 , . .…”

“…Proof. By Theorem 6 of [4] we have rankB L(n,E) = C n n−2 and therefore Π αrs : α rs ∈ I(n − 2, 2n) F is a vector space of dimension C n n−2 . Now let H = (h 1 , h 2 , .…”