Stability of k mod p multisets and small weight codewords of the code generated by the lines of PG(2, q)

Abstract In this paper, we prove a stability result on k mod p multisets of points in PG ( 2 , q ) , q = p h . The particular case k = 0 is used to describe small weight codewords of the code generated by the lines of PG ( 2 , q ) , as linear combination of few lines. Earlier results proved this for codewords with weight less than 2.5q, while our result is valid until c q q . It is sharp when 27 q square and h ≥ 4 . When q is a prime, De Boeck and Vandendriessche (see [2] ) constructed a codeword of weight 3 p − 3 that is not the linear combination of three lines. We characterise their example.