2009/11/10 by Peter Sin, Junhua Wu, Sin, Peter +3
Computer Science · Mathematics · #05E18 #20C20 #94B05 #Coding theory and cryptography #Combinatorics (math.CO) #Cooperative Communication and Network Coding #Error Correcting Code Techniques #FOS: Mathematics #Representation Theory (math.RT) #math.CO #math.RT #msc:05E18 #msc:20C20 #msc:94B05
paper · pdf · doi:10.48550/arxiv.0911.2018
36 pages
arxiv created 2009/11/10 · openalex publication_date 2009/11/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let O be a conic in the classical projective plane PG(2,q), where q is an odd prime power. With respect to O, the lines of PG(2,q) are classified as passant, tangent, and secant lines, and the points of PG(2,q) are classified as internal, absolute and external points. The incidence matrices between the secant/passant lines and the external/internal points were used in \citekeith1 to produce several classes of structured low-density parity-check binary codes. In particular, the authors of \citekeith1 gave conjectured dimension formula for the binary code L which arises as the \Ff2-null space of the incidence matrix between the secant lines and the external points to O. In this paper, we prove the conjecture on the dimension of L by using a combination of techniques from finite geometry and modular representation theory.