Further results on error correcting binary group codes

Two New Four-Error-Correcting Binary Codes

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...

Group Homomorphisms as Error Correcting Codes

We investigate the minimum distance of the error correcting code formed by the homomorphisms between two finite groups G and H. We prove some general structural results on how the distance behaves with respect to natural group operations, such as passing to subgroups and quotients, and taking products. Our main result is a general formula for the distance when G is solvable or H is nilpotent, i...

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...

On the symmetry group of perfect 1-error correcting binary codes

It is shown that for any rank r with n− log(n+1)+4 ≤ r ≤ n−4 and any length n, where n = 2 − 1 and k ≥ 8, there is a perfect code with these parameters and with a trivial group of symmetries.