Minimum weight words of binary codes associated with finite projective geometries

✍ Scribed by Bhaskar Bagchi; N.S. Narasimha Sastry

• Publisher: Elsevier Science
• Year: 1985
• Tongue: English
• Weight: 209 KB
• Volume: 57
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(85)90183-9

✦ Synopsis

Let PG(n, s), s = 2 ~ and n >12, denote the Desarguesian projective space of projective dimension n over the Galois field Fs. The set of its subsets with set theoretic symmetric difference as addition is a vector space over F2. For 1 ~< t n -1, let Ct(n, s) denote its subspace generated by the t-flats of PG(n, s) and for w c_ PG(n, s), let [wl denote the cardinality (or weight) of w. Our object in this note is to present a purely geometric proof of the following theorem proved independently by Smith [5] and Delsarte et al. [2]. Theorem. For s = 2", n > 1 and 0<t<n, the words of Ct(n, s) of least non-zero weight are precisely the t-fiats of PG(n, s). Some crucial parts of the proof are contained in the following lemmas.