## ALGEBRA SEMINAR: On Arcs and MDS Codes

### ALGEBRA SEMINAR: On Arcs and MDS Codes Speaker: Simeon Ball

Title: On arcs and MDS codes

Date/Time:February 7, 2019 / 14:40 - 15:30

Place: FENS G015

Abstract: On Arcs and MDS Codes Simeon Ball Universitat Polit´ecnica Catalunya Barcelona A block code C of length n, minimum distance d over an alphabet with q symbols, satisfies, |C| 6 q n−d+1 , which is known as the Singleton bound. A block code attaining this bound is known as a Maximum Distance Separable code or simply an MDS code. An arc S in F k q is a subset of vectors with the property that every subset of size k of S is a set of linearly independent vectors. Equivalently, an arc is a subset of points of PG(k − 1, q), the (k − 1)-dimensional projective space over Fq, for which every subset of k points spans the whole space. If C is a k-dimensional linear MDS code over Fq then the columns of a generator matrix for C are an arc in F k q and vice-versa. The classical example of a linear MDS code is the Reed-Solomon code, which is the evalutaion code of all polynomials of degree at most k − 1 over Fq. As an arc, the Reed-Solomon code is a normal rational curve in PG(k − 1, q). The trivial upper bound on the length n of a k-dimensional linear MDS code over Fq is n 6 q + k − 1. The (doubly-extended) Reed-Solomon code has length q + 1. The dual of a k-dimensional linear MDS code is a (n − k)-dimensional linear MDS code, thus if we can assume that k 6 1 2 n and therefore that k 6 q − 1. The MDS conjecture states that if 4 6 k 6 q − 2 then an MDS code has length at most q + 1. In other words, there are no better MDS codes than the Reed-Solomon codes. In 2012, the linear MDS conjecture was verified for q prime. In this talk I will talk about various advances since then, survey the non-linear case and highlight the lack of examples of known MDS codes of length more than k + 1 2 q.

Contact: Michel Lavrauw

Faculty of Engineering and Natural Sciences