An empty interval in the spectrum of small weight codewords in the code from points and k-spaces of PG(n, q)

TL;DR

This paper investigates the spectrum of small weight codewords in codes derived from points and k-spaces in projective geometries, establishing the absence of codewords within certain weight intervals and providing bounds on small weight codewords.

Contribution

It introduces new non-existence results for codewords with specific weights in codes from projective geometries, extending previous bounds and covering more general cases.

Findings

No codewords of certain weights exist in specified intervals.

Sharp bounds on small weight codewords are established.

Results generalize previous findings for specific q and n.

Abstract

Let Ck(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 pa- per, we show that there are no codewords of weight in the open interval ] q^{k+1}-1/q-1, 2q^k[ in Ck(n, q) \ Cn-k(n, q) which implies that there are no codewords with this weight in Ck(n, q) \ Ck(n, q) if k >= n/2. In par- ticular, for the code Cn-1(n, q) of points and hyperplanes of PG(n, q), we exclude all codewords in Cn-1(n, q) with weight in the open interval ] q^n-1/q-1, 2q^n-1[. This latter result implies a sharp bound on the weight of small weight codewords of Cn-1(n, q), a result which was previously only known for general dimension for q prime and q = p2, with p prime, p > 11, and in the case n = 2, for q = p^3, p >= 7 ([4],[5],[7],[8]).