On the construction of perfect codes from HEX signal constellations

Recently, minimum and non-minimum delay perfect codes were proposed for any channel of dimension n. Their construction appears in the literature as a subset of cyclic division algebras over Q ( ζ 3 ) only for the dimension n = 2 s n 1 , where s ∈ { 0 , 1 } , n1 is odd and the signal constellations are isomorphic to Z [ ζ 3 ] n . In this work, we propose an innovative methodology to extend the construction of minimum and non-minimum delay perfect codes as a subset of cyclic division algebras over Q ( ζ 3 ) , where the signal constellations are isomorphic to the hexagonal A 2 n - rotated lattice, for any channel of any dimension n such that gcd ( n , 3 ) = 1 .