On Optimal Binary One-Error-Correcting Codes of Lengths $2^{m}-4$ and $2^{m}-3$

On Optimal Binary One-Error-Correcting Codes of Lengths

Best and Brouwer [Discrete Math. 17 (1977), 235– 245] proved that triply-shortened and doubly-shortened binary Hamming codes (which have length 2m − 4 and 2m − 3, respectively) are optimal. Properties of such codes are here studied, determining among other things parameters of certain subcodes. A utilization of these properties makes a computeraided classification of the optimal binary one-erro...

On Optimal Binary One-Error-Correcting Codes of Lengths $2^m-4$ and $2^m-3$

Best and Brouwer [Discrete Math. 17 (1977), 235– 245] proved that triply-shortened and doubly-shortened binary Hamming codes (which have length 2m − 4 and 2m − 3, respectively) are optimal. Properties of such codes are here studied, determining among other things parameters of certain subcodes. A utilization of these properties makes a computeraided classification of the optimal binary one-erro...

One-point Goppa Codes on Some Genus 3 Curves with Applications in Quantum Error-Correcting Codes

We investigate one-point algebraic geometric codes CL(D, G) associated to maximal curves recently characterized by Tafazolian and Torres given by the affine equation yl = f(x), where f(x) is a separable polynomial of degree r relatively prime to l. We mainly focus on the curve y4 = x3 +x and Picard curves given by the equations y3 = x4-x and y3 = x4 -1. As a result, we obtain exact value of min...

Binary cyclic codes from explicit polynomials over GF(2m)

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, monomials and trinomials over finite fields with even characteristic are employed to construct a number of families of binary cyclic codes. Lower bounds on the minimum weight of some fam...

Two New Four-Error-Correcting Binary Codes