Blocking sets in finite projective spaces and uneven binary codes

✍ Scribed by W.Edwin Clark

• Publisher: Elsevier Science
• Year: 1991
• Tongue: English
• Weight: 373 KB
• Volume: 94
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(91)90307-n

✦ Synopsis

Clark, W.E., Blocking sets in finite projective spaces and uneven binary codes, Discrete Mathematics 94 (1991) 65-68. A l-blocking set in the projective space PG(m, 2), m >2, is a set B of points such that any (m -I)-flat meets B and no l-flat is contained in B. A binary linear code is said to be uneuen if it contains at least one codeword of odd weight. If B is a l-blocking set in PG(r -1,2) and dim(B)=r-1 any matrix H whose columns are the vectors in B is a parity check matrix for an uneven binary code of length n = 1B1, redundancy r, and minimum distance at least 4; Conversely, if B is the set of columns of the parity check matrix of such a code then it is a l-blocking set. Using this and results on uneven binary codes of minimum distance 4, the author shows that there exists a l-blocking set of cardinality n if and only if 5 G n G 5 -2"-3.

Beutelspacher [l] defined a t-blocking set of PG(m, q), m 3 t + 1, to be a subset B of PG(m, q) such that any (mt)-flat meets B and no t-flat is contained in B; he proved that such a t-blocking set B satisfies q'+ q'-'+ . . .