Non-Vanishing Determinant Space-Time Codes from Abelian Crossed Product Algebras
Abstract
Every explicit non-cyclic crossed product space--time code for more than four antennas has relied on transcendental structure constants, and its coding gain is therefore not bounded away from zero. We explain this phenomenon and show how to avoid it. Over a number field, the commuting-generator (tensor product) construction yields a division algebra only in the cyclic case, and a division crossed product of prime-power degree requires two places with full decomposition group; in particular, no (Z/2)r, r≥3, crossed product division algebra exists over Q(i). We then introduce Abelian crossed products with a commutator twist, give the general parametric form of their codewords as a 2×2 block matrix of cyclic-algebra codewords, and prove the division property with a centralizer argument and a tame norm-residue symbol. This yields algebraic codes over Q(i) with non-vanishing determinants and closed-form minimum determinants: an information-lossless 4×4 (Z/2)2 code with δmin=1/2000, and an infinite family of DMT-optimal 8×8 Z/4×Z/2 codes indexed by primes p≡5pmod8, whose commutator and cyclic constants are roots of unity. The member p=5, defined over Q(ζ40), has minimum determinant exactly 1/5 before normalization, at least $34.7$ times that of the 8×8 perfect code at equal energy, and its rate-2 subcode for 8×2 systems has at least $5$ times the minimum determinant of the punctured perfect code. A determinant-spectrum analysis shows that at practical error rates the multiplicity of small determinants, rather than their minimum, governs performance, and it explains all simulated results.
Related
- Selfless Reduced Crossed Product C*-Algebras Arising from Almost Periodic Actions
- Continuous family of compact quantum metric space structures from cocycle twisted crossed product C∗-algebras
- Covariant representations of actions of inverse semigroups: a new approach to the reduced and essential crossed-product C*-algebras
- On AF- and type I-ideals in certain crossed product C∗-algebras
- Unique equivariant pseudo expectation and generalized Powers averaging for crossed product
BibTeX
@misc{barzegar2026nonvanishing,
title = {{Non-Vanishing Determinant Space-Time Codes from Abelian Crossed Product Algebras}},
author = {Farhad Barzegar and Dariush Kiani and Hassan Khodaiemehr},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34110},
eprint = {2610.34110},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34110},
}
RIS
TY - UNPB
TI - Non-Vanishing Determinant Space-Time Codes from Abelian Crossed Product Algebras
AU - Farhad Barzegar
AU - Dariush Kiani
AU - Hassan Khodaiemehr
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34110
UR - https://arxiv.org/abs/2610.34110
AB - Every explicit non-cyclic crossed product space--time code for more than four antennas has relied on transcendental structure constants, and its coding gain is therefore not bounded away from zero. We explain this phenomenon and show how to avoid it. Over a number field, the commuting-generator (tensor product) construction yields a division algebra only in the cyclic case, and a division crossed product of prime-power degree requires two places with full decomposition group; in particular, no (Z/2)^r, r≥3, crossed product division algebra exists over Q(i). We then introduce Abelian crossed products with a commutator twist, give the general parametric form of their codewords as a 2×2 block matrix of cyclic-algebra codewords, and prove the division property with a centralizer argument and a tame norm-residue symbol. This yields algebraic codes over Q(i) with non-vanishing determinants and closed-form minimum determinants: an information-lossless 4×4 (Z/2)^2 code with δ_min=1/2000, and an infinite family of DMT-optimal 8×8 Z/4×Z/2 codes indexed by primes p≡5pmod8, whose commutator and cyclic constants are roots of unity. The member p=5, defined over Q(ζ_40), has minimum determinant exactly 1/5 before normalization, at least $34.7$ times that of the 8×8 perfect code at equal energy, and its rate-2 subcode for 8×2 systems has at least $5$ times the minimum determinant of the punctured perfect code. A determinant-spectrum analysis shows that at practical error rates the multiplicity of small determinants, rather than their minimum, governs performance, and it explains all simulated results.
ER -