Subfield-Compatible Polynomials over Finite Fields

Polynomial functions over finite fields are important in computer science and electrical engineering in that they present a mathematical representation of arithmetic circuits. This paper establishes necessary and sufficient conditions for polynomial functions with coefficients in a finite field and naturally restricted degrees to be compatible with given subfields. Most importantly, this is done for the case where the domain and codomain fields have differing cardinalities. These conditions, which are presented for polynomial rings in one and several variables, are developed via a universal permutation that depends only on the cardinalities of the given fields

On the Shape of the General Error Locator Polynomial for Cyclic Codes

General error locator polynomials were introduced in 2005 as an alternative decoding for cyclic codes. We now present a conjecture on their sparsity, which would imply polynomial-time decoding for all cyclic codes. A general result on the explicit form of the general error locator polynomial for all cyclic codes is given, along with several results for specific code families, providing evidence to our conjecture. From these, a theoretical justification of the sparsity of general error locator polynomials is obtained for all binary cyclic codes with t <= 2 and n < 105, as well as for t = 3 and n < 63, except for some cases where the conjectured sparsity is proved by a computer check. Moreover, we summarize all related results, previously published, and we show how they provide further evidence to our conjecture. Finally, we discuss the link between our conjecture and the complexity of bounded-distance decoding of the cyclic codes