A family of low density matrices in Lagrangian-Grassmannian
This paper shows a relationship among the Lagrangian-Grassmannian variety \(\mathrm{LG}(n,2n)\) (which parametrizes the set of maximal isotropic subspaces of a symplectic vector space \(E\)\, of dimension \(2n\))\, defined over a finite field \(\mathbb{F}_q\)\, and the Low Density Parity-Check (LDPC) codes over \(\mathbb{F}_q\). In a previous work the first author of the present paper and \textit{F. Zaldivar} [Des. Codes Cryptography 60, No. 3, 291--298 (2011; Zbl 1225.94028)] proved that, over any field \(\mathbb{F}\), \(\mathrm{LG}(n,2n)=\mathrm{Gr}(n,2n)\cap Z(\Pi_{{\alpha}_{rs}})\),\, with \(\mathrm{Gr}(n,2n)\)\, the Grassmannian variety and \(\Pi_{{\alpha}_{rs}}\)\, a set of of linear homogeneous polynomials. That result is also improved now. Section 2 introduces, over any field \(\mathbb{F}\), the low density Boolean matrices \(\mathcal{L}_k\) and, based on them and according to a certain rule \(\mathcal{R}\), the low density Boolean matrix \(B\). Theorem 6 characterizes, in terms of \(\mathrm{char}(\mathbb{F})\), when those matrices have maximum rank. Section 3 shows that matrices \(\mathcal{L}_k\)\, define regular LDPC codes (Theorem 10) while matrix \(B\)\, define an irregular LDPC code (Theorem 12) (over a finite field \(\mathbb{F}_q\)). Section 4 proves that the family \(\Pi_{{\alpha}_{rs}}\)\, (see above) is unique up to linear combination (Theorem 16). Finally Proposition 16 gives an injective map \([E]\longrightarrow (\mathcal{R}, \mathcal{L}_r,\dots, \mathcal{L}_2)\), \(E\)\, a symplectic space, providing so a matrix invariant for \(E\).
scientific article; zbMATH DE number 6890447