A lower bound on the minimum weight of some geometric codes
Bence Csajbók, Giovanni Longobardi, Giuseppe Marino, Rocco Trombetti
TL;DR
The paper addresses the problem of determining the minimum weight of the duals of incidence-based $p$-ary codes from projective and affine geometries. By viewing dual codewords as $(0 mod p)$-multisets meeting every line and applying a polynomial technique inspired by Ball et al., the authors derive new lower bounds and construct tight examples. They prove a sharp affine bound on the sum of coordinates for $(0 mod p)$-multisets, construct multisets attaining this bound via linear sets and scattered polynomials, and establish a stronger projective bound that improves the Bagchi–Inamdar bound for the case $h>1$ and $m,p>2$, with implications for the minimum weight of ${ m C}_{ ext{PG}}(m,q)^ot$ and related incidence codes. These results deepen the geometric understanding of dual code weights and provide concrete, geometry-based improvements to existing bounds.
Abstract
The $p$-ary code associated with the incidence structure of points and $t$-spaces in a projective space $\mathrm{PG}(m,q)$, where $q=p^h$, is the $\mathbb{F}_p$-subspace generated by the incidence vectors of the blocks of this design. The dual of this code consists of all vectors orthogonal to every codeword of the original code. In contrast to the codes derived from point-subspace incidences, the minimum weight of the corresponding dual codes is generally unknown, which makes the problem more challenging. In 2008 Lavrauw, Storme and Van de Voorde proved the following reduction: the minimum weight of the dual of the code derived from point and $t$-space incidences in $\mathrm{PG}(m,q)$ is the same as the minimum weight of the dual of the code derived from point and line incidences in $\mathrm{PG}(m-t+1,q)$. After a series of works by Delsarte (1970), Assmus and Key (1992), Calkin, Key and De Resmini (1999), the best known lower bound for the case of point-line incidences was established in [B. Bagchi and P. Inamdar: Projective geometric codes, J. Combin. Theory Ser. A, 99(1) (2002), 128-142]. The problem of determining the minimum weight of these codes admits a natural geometric interpretation in terms of multisets of points in a projective space which meet each line in $0$ modulo $p$ points. In this paper, by adopting this geometrical perspective and exploiting certain polynomial techniques from [S. Ball, A. Blokhuis, A. Gács, P. Sziklai, Zs. Weiner: On linear codes whose weights and length have a common divisor, Adv. Math., 211 (2007), 94-104], we prove a substantial improvement of the Bagchi-Inamdar bound in the case where $h>1$ and $m, p >2$.