Small Weight Code Words of Projective Geometric Codes
Abstract
We investigate small weight code words 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 code words of Cj,k(n,q) up to weight (3 - O( 1 q ) ) []0ptk+1j+1q 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 code words (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.
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