Mathematical contributions on the construction of good codes for secret sharing and storage

Vortrag von Prof. Dr. Sihem Mesnager

Sprecher eingeladen von: Prof. Dr. Joachim Rosenthal

Datum: 06.12.17  Zeit: 15.00 - 16.00  Raum: Y27H12

This talk is divided in two parts: the first part is a contribution on the construction of new linear p-ary codes (from plateaued functions in any characteristic) for secret sharing. The second part is a contribution on the construction of new locally recoverable codes (LRC codes) for storage. Below, more details.
Part 1: certain special types of functions over finite fields are closely related to linear or nonlinear codes. In the past decade, a lot of progress on interplays between special functions and codes has been made. Recently, several new approaches to constructing linear codes with special types of functions were proposed, and a lot of linear codes with excellent parameters were obtained. Very recently, some authors have highlighted that bent functions lead to the construction of interesting linear codes. The first part of this talk is devoted to linear codes from plateaued functions in any characteristic. More specifically, we present new linear codes with few weights from weakly regular plateaued functions based on a generic construction.
Part 2: in 2014, a family of optimal linear locally recoverable codes (LRC codes) that attain the maximum possible distance (given code length, cardinality, and locality) is presented by Tamo and Barg. The key ingredient for constructing such optimal linear LRC codes is the so-called r-good polynomials, where r is equal to the locality of the LRC code. However, given a prime p, known constructions of r-good polynomials on some extension field of GF p exist only for some special integers r, and the problem of constructing optimal LRC codes over small field for any given locality is still open. By using function composition, we present in the second part of this talk two general methods of designing good polynomials, which lead to three new constructions of r-good polynomials. Such polynomials bring new constructions of optimal LRC codes. In particular, our constructed polynomials as well as the power functions yield optimal (n,k,r)-LRC codes over GF q for all positive integers r as localities, where q is near the code length n.