Some good linear codes from functions over finite fields and their applications
This talk is divided in two parts: the first part is a contribution on the construction of new linear p-ary codes (from bent functions and plateaued functions in any characteristic) for secret sharing and two-party computation. The second part is a contribution on the construction of new locally recoverable codes (LRC codes) for storage. Below, more details.
Part 1: The first part of this talk is devoted to minimal linear codes from bent/plateaued functions in any characteristic. We will present two generic constructions of linear codes involving special functions and investigate constructions of good linear codes based on the generic constructions involving bent and plateaued functions over finite fields. More specifically, we present new minimal linear codes with few weights from weakly regular bent/plateaued functions based on generic constructions.
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-1 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. We present in the second part of this talk general methods of designing good polynomials, which lead to new constructions of r-good polynomials. Such polynomials bring new constructions of optimal LRC codes.