Association schemes and coding theory

Uniqueness of the 120-point spherical 11-design in four dimensions

“…Another interesting property of the 600-cell is that it is the only known maximal spherical code in n м 4 dimensions whose maximality does not follow by the Levenshtein bound [16][17][18] (see also [12]) neither by the linear programming improvements of the Levenshtein bounds [9]. Indeed, the relevant Levenshtein bound gives A(4, cos(π/5)) Ϲ 121.67 and our program SCOD (see, e.g.…”

“…The next two theorems [11, Theorems 4.3 and 5.10] (see also [18,12]) are known as the linear programming bounds for spherical designs and for spherical codes, respectively.…”

On the Automorphism Groups of the $$\mathbb{Z}_{2}\mathbb{Z}_{4}$$ -Linear Hadamard Codes and Their Classification

“…Any nonempty subset C of Z Additive codes were first defined by Delsarte in 1973 as subgroups of the underlying Abelian group in a translation association scheme [7,8]. In the special case of a binary Hamming scheme, that is, when the underlying Abelian group is of order 2 n , the additive codes coincide with the codes that are subgroups of Z2 Z4 .…”

Maximum distance separable codes over $${\mathbb{Z}_4}$$ and $${\mathbb{Z}_2 \times \mathbb{Z}_4}$$

“…For the binary Hamming scheme, the only structures for the abelian group are those of the form Z α 2 × Z β 4 , with α + 2β = n [4]. Thus, the subgroups C of Z α 2 × Z β 4 are the only additive codes in a binary Hamming scheme.…”