Talk
A Finite Geometry Construction for MDPC-Codes
Talk by Jessica Bariffi
Date: 18.03.20 Time: 16.15 - 17.15 Room:
This master thesis presents a new construction of MDPC-codes, which are codes characterized by a parity-check matrix that is of row-weight \(\mathcal{O}(\sqrt{n})\). We provide a construction of a parity-check matrix using finite geometry. To be precise, we work in the desarguesian projective plane \(PG(2,q)\) of order \(q\), where \(q\) is an odd prime power. The partiy-check matrix we present is the concatenation of two matrices. The first one is an incidence matrix of points and lines in \(PG(2,q)\). For the second matrix, we use projective bundles in \(PG(2,q)\). Projective bundles are a collection of \(q^2+q+1\) non-degenerate conics in \(PG(2,q)\) that mutually intersect in a unique point of the desarguesian projective plane. These conics can then be interpreted as lines in \(PG(2,q)\) and therefore the incidence structure consisting of the set of points of \(PG(2,q)\) and a projective bundle of \(PG(2,q)\) is a projective plane. Therefore we choose the second matrix to be an incidence matrix of points and conics in a projective bundle in \(PG(2,q)\). The parity-check matrices constructed in that way characterize the family of MDPC-codes of length \(2(q^2+q+1)\) and dimension \(q^2+q\). Furthermore the minimum distance of such an MDPC-code is at least \(\left\lceil \frac{2q+4}{3} \right\rceil\). By the properties of lines and non-degenerate conics of a projective bundles in the desarguesian plane, the maximum column intersection can be reduced to 2, which affects that within one round of Gallager's bit-flipping algorithm applied on a parity-check matrix of such an MDPC-codes we are able to correct \(\lfloor\frac{q+1}{4}\rfloor\) errors.