An Elementary Approach to Ax-Katz, McEliece's Divisibility and Applications to Quasi-Perfect Binary 2-Error Correcting Codes

Abstract

Abstract-In this paper we present an algorithmic approach to the problem of the divisibility of the number of solutions to a system of polynomial equations. Using this method we prove that all binary cyclic codes with two zeros over F 2 f and minimum distance 5 are quasi-perfect for f ≤ 10. We also present elementary proofs of divisibility results that, in some cases, improve previous results.

On the covering radius of some binary cyclic codes

“…When the minimum distance of the code in previous results is 5 we obtain a quasi-perfect code. In [5], Moreno et al stated the following conjecture: Prove that all the binary primitive cyclic codes with two zeros and minimum distance 5 are quasi-perfect.…”