Fast-decodable MIDO codes from non-associative algebras

Abstract

A note on versions:The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the repository url above for details on accessing the published version and note that access may require a subscription.For more information, please contact eprints@nottingham.ac.uk FAST-DECODABLE MIDO CODES FROM NONASSOCIATIVE ALGEBRAS S. PUMPLÜN AND A. STEELEAbstract. By defining a multiplication on a direct sum of n copies of a given cyclic division algebra, we obtain new unital nonassociative 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 generalizes the iterated codes in [20].

Nonassociative cyclic extensions of fields and central simple algebras

“…We point out that nonassociative generalized cyclic algebras have been recently successfully used both in constructing space-time block codes and linear codes [19,20,21,22].…”

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

“…If S is a division algebra, the associative quotient algebras S[t; σ, δ]/( f ) as well as the right nuclei of the nonassociative algebras S f were used when constructing central simple algebras for instance in [1,2,28], [29, Sections 1.5, 1.8, 1.9], [42]. Due to their large nuclei, the algebras S f were also successfully employed to systematically build fast-decodable fully diverse space-time block codes in [37,48,54], see [49], which are used for reliable high rate transmission over wireless digital channels with multiple antennas transmitting and receiving the data. Skew-polynomial rings and their ideals have been already used in other applications and when generalizing other classical notions like Gröbner bases [3] to a non-commutative setting, e.g.…”

“…Throughout the paper we will put a special emphasis on the nonassociative cyclic algebras (K /F, σ, c) employed in [55], and on the generalized nonassociative cyclic algebras (D, σ, d), since these are used for iterated space-time block codes [48,49].…”

“…For m = 2, γ (A) is used in the iterated codes constructed in [37]. In particular, for d ∈ F × the algebra in Example 2 is employed for the space-time block codes in [54], see also [48].…”

“…The multiplication of the division algebra A = (D, τ, ω) is behind the fully diverse codes employed in [54] (cf. [48]). Here,…”