Small weight codewords of projective geometric codes

Abstract

We investigate small weight codewords of the p-ary linear code Cj,k(n,q) generated by the incidence matrix of k-spaces and j-spaces of PG(n,q) and its dual, with q a prime power and 0⩽j<k<n. Firstly, we prove that all codewords of Cj,k(n,q) up to weight (3−O(1q))[k+1j+1]q are linear combinations of at most two k-spaces (i.e. two rows of the incidence matrix). As for the dual code Cj,k(n,q)⊥, we manage to reduce both problems of determining its minimum weight (1) and characterising its minimum weight codewords (2) to the case C0,1(n,q)⊥. This implies the solution to both problem (1) and (2) if q is prime and the solution to problem (1) if q is even.