Rati Ludhani: Minimal Codewords of Projective Reed-Muller Codes of Order 2

Minimal codewords of a linear code reveal its important structural properties and are required, for instance, in secret-sharing schemes and certain decoding algorithms. A nonzero codeword is said to be minimal if its support does not properly contain the support of any other nonzero codeword. Determining minimal codewords of a general linear code is NP-hard, so one typically exploits the specific structure of a given code. Here, we consider this problem for projective Reed Muller (PRM) codes of order 2.

PRM codes of order 2 are evaluation codes obtained by evaluating quadratic forms over a finite field F at the F-rational points of the corresponding projective space. To characterize their minimal codewords, we reduce the problem to the following geometric question: given two quadrics such that the F-rational points of one are contained in the other, can they differ? Our main result is that for absolutely irreducible quadrics, this almost never happens. In this talk, we present a complete answer to this question, thereby classifying the minimal codewords of PRM codes of order 2.

This is joint work with Alain Couvreur.