The covering radius of long primitive ternary BCH codes

Franken, Ralf (2005) The covering radius of long primitive ternary BCH codes. PhD thesis, University of Glasgow.

Abstract

This thesis is about the generalisation of a method to determine an asymptotic upper bound for the covering radius of primitive BCH codes. The method was introduced by S. D. Cohen in the mid-1990s for binary codes. It reduces the coding-theoretical problem to the complete splitting of a single polynomial F(x) over a finite field, which is then established using results that have their roots in ramification theory of function fields. The opening chapter introduces the covering radius problem for BCH codes along with its full coding-theoretical background and some history. As a first result, the transformation from the covering radius problem to a polynomial splitting problem is extended to primitive p-ary BCH codes, where p is an arbitrary prime. The process, during which an explicit "ready-to-use" form of the general F is derived, is summarised in one theorem (Theorem 6).The foundations for arranging the splitting of F (via certain adjustable coefficients) were laid in previous work by Cohen, which is presented in extracts. By combining the key strategy of this with new ideas to meet the special requirements of the non-binary case, sufficient criteria for the splitting are obtained; these come in the form of conditions on polynomials ƒ[0] and ƒ[1], where F has been parameterised as ƒ[0] + uƒ[1] (u an indeterminate). Several other lemmas are proved to deal with the establishing of the conditions. All these results are valid for arbitrary primes p ≥ 3, so that with this the desired general version of the method has been made available.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Additional Information: Adviser: S D Cohen
• Keywords: Theoretical mathematics
• Date of Award: 2005
• Depositing User: Enlighten Team
• Unique ID: glathesis:2005-74054
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 23 Sep 2019 15:33
• Last Modified: 23 Sep 2019 15:33
• URI: https://theses.gla.ac.uk/id/eprint/74054