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Non-Vanishing Determinant Space-Time Codes from Abelian Crossed Product Algebras

Farhad Barzegar, Dariush Kiani, Hassan Khodaiemehr

arXiv:2610.34110v1cs.ITmath.ITmath.RA

TL;DR

Earlier explicit non-cyclic crossed-product codes for more than four antennas relied on transcendental constants, whose determinants lack a positive lower bound. The paper introduces commutator-twisted Abelian crossed products, gives explicit block codewords, and proves division using a centralizer criterion. C4 beats the 4 × 4 perfect code by about 0.4–0.5 dB under exact ML; Cu8 beats the 8 × 8 perfect code by 0.3–0.7 dB at BLER 10−2–10−3 under near-ML, while C8’s advantage is asymptotic.

  • Problem

    Earlier explicit non-cyclic crossed-product codes for more than four antennas relied on transcendental constants, whose determinants lack a positive lower bound.

    Source: §I · p. 1

  • Method

    The paper introduces commutator-twisted Abelian crossed products, gives explicit block codewords, and proves division using a centralizer criterion.

    Source: §I · p. 1, C. A computable division criterion · p. 4

  • Results

    C4 beats the 4 × 4 perfect code by about 0.4–0.5 dB under exact ML; Cu8 beats the 8 × 8 perfect code by 0.3–0.7 dB at BLER 10−2–10−3 under near-ML, while C8’s advantage is asymptotic.

    Source: F. Summary · pp. 11–12

  • Takeaways & Limitations

    Determinant-spectrum analysis explains why small-determinant multiplicities can govern practical error rates even when minimum determinants predict asymptotic advantages.

    Source: §X · p. 12

Abstract

from arXiv · show

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 $(\Z/2)^r$, $r\ge3$, 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 $2\times2$ 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 $4\times4$ $(\Z/2)^2$ code with $\delta_{\min}=1/2000$, and an infinite family of DMT-optimal $8\times8$ $\Z/4\times\Z/2$ codes indexed by primes $p\equiv5\pmod8$, whose commutator and cyclic constants are roots of unity. The member $p=5$, defined over $\Q(\zeta_{40})$, has minimum determinant exactly $1/5$ before normalization, at least $34.7$ times that of the $8\times8$ perfect code at equal energy, and its rate-2 subcode for $8\times2$ 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.

I. INTRODUCTION

Earlier non-cyclic crossed-product codes with transcendental constants lack a positive coding-gain floor, while algebraic constants obstruct the commuting construction. The paper develops a twisted construction and explicit algebraic codes, then evaluates their determinants and error performance.

I. INTRODUCTION

  • Transcendental structure constants make division easy but leave determinants without a positive lower bound, motivating algebraic non-cyclic constructions.The paper notes that earlier explicit non-cyclic codes for larger degrees relied on transcendental constants.Source: §I · p. 1

A. Contributions

  • Over number fields, algebraic commuting-generator constructions yield division algebras only in the cyclic case, and (Z/2)^r division crossed products for r ≥ 3 do not exist over Q(i).The paper identifies these obstructions and the admissible Galois groups over Q(i).Source: A. Contributions · p. 1
  • The paper introduces a commutator-twisted Abelian crossed product with an explicit 2 × 2 block codeword form and a computable division criterion.It constructs an information-lossless 4 × 4 code with δmin = 1/2000 and an infinite family of DMT-optimal 8 × 8 codes.Source: A. Contributions · p. 2

II. SYSTEM MODEL AND DESIGN CRITERIA

  • The system model is a quasi-static flat-fading MIMO channel, with codewords energy-normalized and NVD defined by a positive minimum determinant over the infinite code.Information losslessness is established separately for each code rather than assumed for all crossed-product codes.Source: §II · p. 2

III. ALGEBRAIC BACKGROUND

The algebraic background connects division algebras to full diversity in matrix codes. The section then proves that algebraic commuting-generator constructions cannot produce non-cyclic division algebras over number fields, explaining the role and cost of transcendental constants.

III. ALGEBRAIC BACKGROUND

  • A division algebra gives fully diverse codes through its regular representation, whose matrix determinants equal reduced norms.A crossed product is a central simple algebra containing its Galois extension as a maximal subfield.Source: §III · p. 3

A. Commuting generators

  • For algebraic constants over a number field, the commuting-generator construction is a division algebra only when the degrees are pairwise coprime, making the Galois group cyclic.The argument bounds the algebra's exponent and index by the least common multiple of the component degrees.Source: A. Commuting generators · p. 3
  • Transcendental constants evade this obstruction, but their dense determinant subring yields no positive lower bound on the minimum determinant.For almost every unit-circle parameter in the degree-two example, the infimum of the reduced norms is zero.Source: A. Commuting generators · p. 3
  • Among 4-QAM differences, the cited transcendental 4 × 4 code has |det|² ≈ 8.4 × 10^-7, versus 0.128 for C4.This numerical example illustrates the determinant-floor problem for the earlier construction.Source: A. Commuting generators · p. 3

B. Local degrees

Prime-power-degree division crossed products require two places with full decomposition group, ruling out several groups over Q(i). The paper gives a centralizer-based division criterion and introduces a commutator twist that enables explicit codeword forms.

B. Local degrees

  • A prime-power-degree division crossed product requires two distinct finite places with full decomposition group, except for the stated degree-two real-place exception.Consequently, no (Z/2)^r division crossed product for r ≥ 3 exists over Q(i); a (Z/2)^2 case requires two places of full local degree four.Source: B. Local degrees · pp. 3–4
  • The Z/4 × Z/2 group is admissible and can occur at a tame place with ramification index 4 and residue degree 2 when the residue field has q ≡ 1 (mod 4).This observation guides the construction of C8.Source: B. Local degrees · p. 4

C. A computable division criterion

  • The division criterion tests whether a centralizer has a local invariant of order |H| at a place with full decomposition group.Under the stated subgroup condition, that order implies the original algebra has index equal to its degree.Source: C. A computable division criterion · p. 4
  • For a cyclic, totally and tamely ramified local extension, the cyclic algebra's Brauer-class order is computed from the residue-field image of its unit parameter.The criterion uses the multiplicative order of ā^((q−1)/m).Source: C. A computable division criterion · p. 4

V. ABELIAN CROSSED PRODUCT CODES WITH A COMMUTATOR TWIST

  • The twisted relations fe = efu produce closed-form codewords, and the general construction arranges cyclic-algebra codewords in a 2 × 2 block matrix.The twist is necessary because non-cyclic algebraic crossed-product generators cannot commute.Source: §V · p. 4

A. The algebra

  • Setting u = 1 recovers the commuting construction, which never gives a division algebra over a number field when m is even.The relations define an associative crossed product only under the stated cocycle conditions.Source: A. The algebra · p. 4

B. General parametric form of the codewords

The construction gives an explicit matrix form for twisted Abelian crossed-product codewords by organizing their coefficients into cyclic-algebra blocks. This structure makes the coupling and full-rate symbol placement transparent.

B. General parametric form of the codewords

  • The entry formula specifies how each coefficient contributes to the left regular matrix at row (i′, j′) and column (k, l).Its indices satisfy i ≡ i′ − k (mod m) and j ≡ j′ − l (mod 2).Source: B. General parametric form of the… · p. 4
  • Grouping coefficients into even and odd layers reveals a transparent structure built from familiar cyclic-algebra codewords.The cyclic block Cγ(y) is the codeword of the cyclic algebra (L/Lσ, σ, γ).Source: B. General parametric form of the… · pp. 4–5
  • The construction carries n² QAM symbols and is full-rate.The codeword symbols are formed from the lattice basis in the displayed construction.Source: B. General parametric form of the… · p. 5
  • The twisted Z/m×Z/2 codeword arranges four cyclic codewords in a 2×2 block matrix, coupled by u through Du and by b.The block formulas follow by splitting the entries according to the two values of j′ and l.Source: B. General parametric form of the… · p. 5
  • When u = 1 and a ∈ K, the blocks decouple into a Kronecker structure that is never a division algebra over a number field.The passage attributes the nondivision claim to Theorem 2.Source: B. General parametric form of the… · p. 5

C. Shaping and determinants

The section establishes shaping and determinant bounds, describes a lower-cost encoder, and specifies the 4×4 code C4. Its construction is information-lossless and has cubic shaping.

C. Shaping and determinants

  • A unitary-up-to-scale embedding matrix and unit-modulus constants make the code information-lossless.The generator is unitary up to scale under these conditions.Source: C. Shaping and determinants · p. 5
  • The determinant bound gives |nrd(x)|² ≥ N(p)^−⌊n/(2e)⌋ under the stated valuation conditions on b.The bound assumes a and u are units, b has valuation at least −1 above p, and b is integral elsewhere.Source: C. Shaping and determinants · p. 5

D. Encoding

  • For n = 8, the encoder uses 448 complex multiplications, versus 4096 for a generic linear-dispersion code and 576 for a cyclic code with a dense layer matrix.Each codeword entry depends on exactly one layer, but worst-case exhaustive ML decoding remains |S|^n².Source: D. Encoding · p. 5

VI. THE 4 × 4 CODE C4

  • C4 uses m = 2 and a field with two odd primes having e = f = 2; its chosen basis gives information-losslessness and cubic shaping.The layer basis uses the Golden code ideal, and the constants have unit modulus under every embedding.Source: §VI · p. 5, §5 · p. 6

5. We take

  • C4’s explicit codewords carry 16 QAM symbols, with coefficients assigned to 1, e, f, and ef.The codeword form follows from Theorem 8.Source: §5 · p. 6

B. Division property and minimum determinant

The section proves that the degree-four algebra is division and derives an exact minimum determinant for C4. It also compares that determinant with the 4×4 perfect code.

B. Division property and minimum determinant

VII. THE 8 × 8 CODES

For eight antennas, the construction uses the admissible group Z/4×Z/2 and yields a division algebra with an exact determinant bound. The p = 5 code outperforms the 8×8 perfect code in the reported determinant comparison.

VII. THE 8 × 8 CODES

  • For eight antennas, Corollary 4 excludes (Z/2)^3 and leaves Z/4×Z/2, whose commutator and cyclic constants can be roots of unity.The p = 5 instance is defined over Q(ζ40); its constants u and a are roots of unity.Source: §VII · p. 6, A. The code C8 and its explicit… · p. 6

A. The code C8 and its explicit codewords

B. Division property and minimum determinant

5. The corresponding real

The construction trades information losslessness for a denser A4-type layer lattice, while a prime-indexed family provides NVD division codes whose best member is p = 5.

5. The corresponding real

  • The A4-type layer lattice is denser than Z8, but this relaxed-shaping trade-off costs information losslessness.A unitary variant restores information losslessness, with δE ≈ 1/(5.12 × 10^9), numerical, 0.3 dB below the perfect code.Source: §5 · p. 7

C. An infinite family generalizing C8

  • For primes p ≡ 5 (mod 8), the family construction gives degree-8 division algebras and NVD codes with |nrd(x)|2 ≥ 1/p.The construction uses the stated splitting, inertia, and cyclic-quartic-subfield properties of 5; p = 5 gives C8.Source: C. An infinite family generalizing C8 · p. 7
  • As p increases, the energy-normalized minimum determinant decreases rapidly because Gaussian periods grow in norm and the layer lattice has condition number √p.The paper identifies p = 5 as the optimal family member, justifying C8.Source: C. An infinite family generalizing C8 · pp. 7–8

D. Rate-2 puncturing for 8 × 2 systems

Puncturing C8 yields a rate-2 code for 8 × 2 systems with a proved determinant advantage over the punctured perfect code. The resulting NVD codes achieve DMT optimality, and decoding uses exact ML or K-best search depending on dimension and SNR.

D. Rate-2 puncturing for 8 × 2 systems

  • Keeping layers {1, e} gives 16-QAM symbols per 8 channel uses and block-diagonal codewords formed from two 4 × 4 cyclic codewords linked by τ.Even-layer subcodes remain fully diverse and make C8 a mother code for rate-1 to rate-4 asymmetric codes with a common decoder structure.Source: D. Rate-2 puncturing for 8 × 2 systems · p. 8
  • The rate-2 C8 subcode has minimum |det Mx|2 = 1 exactly and at least five times the minimum determinant of the punctured perfect code.The determinant ratio corresponds to 0.87 dB more asymptotic coding gain.Source: D. Rate-2 puncturing for 8 × 2 systems · p. 8

E. DMT optimality

  • C4, C8, Cu8, and every member of the family achieve the optimal diversity–multiplexing tradeoff for every Nr ≥ 1.The proof uses their full-rate NVD property.Source: E. DMT optimality · p. 8

VIII. DECODING AND DETERMINANT-SPECTRUM ANALYSIS

  • The determinant-spectrum analysis examines codeword differences that dominate error probability and is intended to predict the later simulations.Sorted-QR ordering reorders columns for each channel realization, making visited-node counts essentially insensitive to initial vectorization.Source: §VIII · p. 8, A. Maximum-likelihood decoding · p. 8

A. Maximum-likelihood decoding

  • For n = 8 and Nr = 8, exact ML decoding is impractical at low and moderate SNR, so the study uses K-best decoding with K = 1024.For other reported cases, the Schnorr–Euchner sphere decoder with sorted-QR ordering performs exact ML; code rankings are reported as independent of K.Source: A. Maximum-likelihood decoding · p. 8

B. Determinant spectrum and union bound

The determinant-spectrum analysis predicts that small-determinant multiplicity, rather than the minimum determinant alone, governs practical-range comparisons. Simulations report gains for C4 in correlated fading and confirm the transcendental code’s disadvantage as SNR rises.

B. Determinant spectrum and union bound

  • The reported determinant-spectrum computation covers differences of Hamming weight at most 3 for 4 × 4 and 8 × 2 codes, and at most 2 for 8 × 8 codes.Table II reports the smallest determinant d1 and weighted multiplicities A1 and A4; Table III’s searched minima are upper bounds on the true minimum.Source: B. Determinant spectrum and union bound · p. 8, Table III · p. 9
  • C4 has 3.2 times fewer near-minimal differences than the perfect code, with A4 = 496 versus 1580, despite a 1.8-times smaller d1.The analysis predicts moderate-error-rate outperformance despite a 0.62 dB asymptotic deficit.Source: B. Determinant spectrum and union bound · pp. 8–9
  • The transcendental codes have extremely small d1 but A4 = 16, predicting similarity at moderate SNR and falling behind only at high SNR.The union-bound estimate reproduces code ordering and predicts this high-SNR decline.Source: B. Determinant spectrum and union bound · p. 9, Figure 1 · p. 9
  • C8 and its 8 × 2 subcode have minimum determinants larger by factors of 34.7 and 5, respectively, but larger A4; their predicted crossings occur only near 26 dB and 36 dB.At practical error rates, multiplicity dominates, while the larger minimum determinant acts as an asymptotic guarantee; optimizing A4 is left open.Source: B. Determinant spectrum and union bound · p. 9

A. Minimum determinants and information loss

The determinant comparison gives C8 a larger proved minimum than the best value found for the 8×8 perfect code, but simulations show that small-determinant multiplicity better predicts practical error performance. The 4×4 and 8×8 comparisons also show that code performance depends on channel conditions and decoding regime, while the algebraic structure does not reduce ML decoding complexity.

A. Minimum determinants and information loss

  • Among the 8×8 codes, C8's proved determinant is 34.7 times the best value found for the perfect code; Cu8 lies between them.Earlier transcendental codes have determinants several orders of magnitude below every algebraic code, even within the 4-QAM codebook.Source: A. Minimum determinants and… · p. 9

B. Error performance of the 4 × 4 codes (exact ML)

C. Decoding complexity

  • In correlated fading, C4 requires about 35% more sphere-decoder nodes than the perfect code; the algebraic structure does not reduce ML decoding complexity.At 12 dB, C4 visits 5.0·10^4 nodes versus 3.7·10^4 for the perfect code.Source: C. Decoding complexity · pp. 10–11

D. Error performance of the 8 × 8 codes (near ML)

E. Error performance of the 8 × 2 codes (exact ML)

The rate-2 subcode remains slightly behind the punctured perfect code in the simulated range, despite a larger minimum determinant. The paper’s summary and conclusion emphasize that practical performance tracks low-determinant multiplicity, while several design and construction questions remain open.

E. Error performance of the 8 × 2 codes (exact ML)

  • C×2 8 remains 0.18–0.23 dB behind the punctured perfect code between BLER 10^-2 and 10^-4; at 16 dB their BLERs are 1.3·10^-5 and 8.0·10^-6.Its fivefold larger minimum determinant is offset by greater low-determinant multiplicity, and the predicted crossing is below the simulable range.Source: E. Error performance of the 8 × 2… · p. 11

F. Summary

  • At practical error rates, C4 beats the 4×4 perfect code by 0.4–0.5 dB under exact ML, and Cu8 beats the 8×8 perfect code by 0.3–0.7 dB under near-ML decoding.The asymptotic advantages of C8 and C×2 8 are not visible down to BLER 2.5·10^-4 and 8·10^-6, respectively.Source: F. Summary · pp. 11–12

X. CONCLUSION

  • The conclusion identifies optimizing A4, constructing NVD Z/2^k×Z/2 codes for n=16, fast-decodable punctured-code orderings, and discriminant-optimal orders as open problems.The determinant-spectrum analysis links practical performance to low-determinant multiplicity rather than minimum determinant alone.Source: §X · p. 12
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