How to obtain division algebras used for fast-decodable space-time block codes

We present families of unital algebras obtained through a doubling process from a cyclic central simple algebra D, employing a K-automorphism tau and an invertible element d in D. These algebras appear in the construction of iterated space-time block codes. We give conditions when these iterated algebras are division which can be used to construct fully diverse iterated codes. We also briefly look at algebras (and codes) obtained from variations of this method

Fast-decodable MIDO codes from non-associative algebras

By defining a multiplication on a direct sum of n copies of a given cyclic division algebra, we obtain new unital non-associative algebras. We employ their left multiplication to construct rate-n and rate-2 fully diverse fast ML-decodable space-time block codes for a Multiple-Input-Double-Output (MIDO) system. We give examples of fully diverse rate-2 4×2, 6×2, 8×2 and 12×2 space-time block codes and of a rate-3 6×2 code. All are fast ML-decodable. Our approach generalises the iterated codes in Markin and Oggier

Codes as fractals and noncommutative spaces

We consider the CSS algorithm relating self-orthogonal classical linear codes to q-ary quantum stabilizer codes and we show that to such a pair of a classical and a quantum code one can associate geometric spaces constructed using methods from noncommutative geometry, arising from rational noncommutative tori and finite abelian group actions on Cuntz algebras and fractals associated to the classical codes.Comment: 18 pages LaTeX, one png figur

Tensor products of nonassociative cyclic algebras

We study the tensor product of an associative and a nonassociative cyclic algebra. The condition for the tensor product to be a division algebra equals the classical one for the tensor product of two associative cyclic algebras by Albert or Jacobson, if the base field contains a suitable root of unity. Stronger conditions are obtained in special cases. Applications to space–time block coding are discussed

How to obtain lattices from (f,σ,δ)-codes via a generalization of Construction A

We show how cyclic (f,σ,δ)-codes over finite rings canonically induce a Z-lattice in RN by using certain quotients of orders in nonassociative division algebras defined using the skew polynomial f. This construction generalizes the one using certain σ-constacyclic codes by Ducoat and Oggier, which used quotients of orders in non-commutative associative division algebras defined by f, and can be viewed as a generalization of the classical Construction A for lattices from linear codes. It has the potential to be applied to coset coding, in particular to wire-tap coding. Previous results by Ducoat and Oggier are obtained as special cases