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SEMINAR:Additive Maximum Distance Separable Codes24-11-2020

Speaker: Simeon Ball

Title:  Additive Maximum Distance Separable Codes

Date/Time: 25 November 2020/ 13:40 - 14:30

Zoom: Meeting ID 962 6434 3854

Passcode:  algebra

Abstract:Let A be a finite set and let n and k be a positive integers. An MDS code C is a subset of A^n of size |A|^k in which any two elements of C differ in at least n − k + 1 coordinates. In other words, the minimum (Hamming) distance d between any two elements of C is n − k + 1.If A is an abelian group then we define an additive code to be a code C with the property that for all u, v ∈ C, we have u + v ∈ C. If A is a finite field then C is linear over some subfield of A, so we take A = F_q^h and assume that C is linear over F_q. The trivial upper bound on the length n of a k-dimensional additive MDS code over F_q^h is n ≤ q^h + k − 1. The classical example of an MDS code is the Reed-Solomon code, which is the evaluation code of all polynomials of degree at most k − 1 over F_q^h .The Reed-Solomon code is linear over F_q^h and has length q^h + 1. The MDS conjecture states (excepting two specific cases) that an MDS code has length at most q^h + 1. In other words, there are no better MDS codes than the Reed-Solomon codes. We use geometrical and computational techniques to classify all additive MDS codes over F_q^h for q^h ∈ {4, 8, 9}. We also classify the longest additive MDS codes over F_16 which are linear over F_4. These classifications not only verify the MDS conjecture for additive codes in these cases, but also confirm there are no additive non-linear MDS codes which perform as well as their linear counterparts. In this talk I will cover the main geometrical theorem that allows us to obtain this classification and compare these classifications with the classifi-cations of all MDS codes of alphabets of size at most 8, obtained previously by Alderson (2006), Kokkala, Krotov and Osterg ̊ard (2015) and Kokkala and Osterg ̊ard (2016).

Bio:Senior lecturer in the Department of Mathematics at Universitat Politècnica de Catalunya, Barcelona since 2007. Published over 60 articles and been awarded various grants, including the Advanced Research Fellowship from EPSRC in the UK and the Ramon y Cajal grant in Spain. In 2012, proved the MDS conjecture for prime fields, which conjectures that all linear codes over prime fields that meet the Singleton bound are short. This is one of the oldest conjectures in the theory of error-correcting codes. In 2015, published the book entitled “Finite Geometry and Combinatorial Applications” in the London Mathematical Society Student Text series of Cambridge University Press and in 2020, published the book entitled “A Course in Algebraic Error-Correcting Codes” in the Compact Textbook in Mathematics series of Birkhauser.