An empty interval in the spectrum of small weight codewords in the code from points and k-spaces of [formula omitted

Abstract

Let C k ( n , q ) be the p-ary linear code defined by the incidence matrix of points and k-spaces in PG ( n , q ) , q = p h , p prime, h ⩾ 1 . In this paper, we show that there are no codewords of weight in the open interval ] q k + 1 − 1 q − 1 , 2 q k [ in C k ( n , q ) ∖ C n − k ( n , q ) ⊥ which implies that there are no codewords with this weight in C k ( n , q ) ∖ C k ( n , q ) ⊥ if k ⩾ n / 2 . In particular, for the code C n − 1 ( n , q ) of points and hyperplanes of PG ( n , q ) , we exclude all codewords in C n − 1 ( n , q ) with weight in the open interval ] q n − 1 q − 1 , 2 q n − 1 [ . This latter result implies a sharp bound on the weight of small weight codewords of C n − 1 ( n , q ) , a result which was previously only known for general dimension for q prime and q = p 2 , with p prime, p > 11 , and in the case n = 2 , for q = p 3 , p ⩾ 7 [K. Chouinard, On weight distributions of codes of planes of order 9, Ars Combin. 63 (2002) 3–13; V. Fack, Sz.L. Fancsali, L. Storme, G. Van de Voorde, J. Winne, Small weight codewords in the codes arising from Desarguesian projective planes, Des. Codes Cryptogr. 46 (2008) 25–43; M. Lavrauw, L. Storme, G. Van de Voorde, On the code generated by the incidence matrix of points and hyperplanes in PG ( n , q ) and its dual, Des. Codes Cryptogr. 48 (2008) 231–245; M. Lavrauw, L. Storme, G. Van de Voorde, On the code generated by the incidence matrix of points and k-spaces in PG ( n , q ) and its dual, Finite Fields Appl. 14 (2008) 1020–1038].