Finite Codebook Complexity Overview

• Finite Codebook Complexity is a framework that defines the design, evaluation, and optimization of finite sets of vectors for robust communications, sensing, and quantum information.
• It employs algebraic methods such as generalized Jacobi sums, trace functions, and ring-based characters to achieve near-optimal cross-correlation bounds and parameter flexibility.
• The study highlights significant computational challenges, emphasizing the need for approximate, scalable methods in practical high-throughput and distributed systems.

Finite Codebook Complexity (FC Complexity) formalizes the trade-offs and algorithmic challenges in designing, evaluating, and optimizing codebooks—finite sets of complex-valued or real-valued vectors—for applications in communications, distributed computation, sensing, and quantum information. Core concerns include minimizing cross-correlation amplitudes, achieving optimal packing (e.g., reaching the Welch or Levenshtein bound), controlling parameter growth with alphabet constraints, and quantifying the computational hardness of constructing or decoding near-optimal codebooks in both blocklength and correlation regimes.

1. Structural and Algebraic Principles in Codebook Design

A foundational strategy in codebook construction is the algebraic exploitation of group and field structures. Classical and generalized character sums—Jacobi, Gauss, and newly defined hybrid variants—serve as building blocks for codewords with prescribed cross-correlation profiles. For instance, codebooks can be built via vectors whose coordinates are parameterized by multiplicative or additive characters over finite fields, local rings, or structured vector spaces. Many such constructions feature highly regular inner-product distributions, which are essential for applications requiring equiangular tight frames or near-optimal line packings.

Key algebraic techniques include:

• Generalized Jacobi Sums: These are formed over tuples of finite field elements, using trace conditions and variable field extensions to generalize prior constructions. This yields a broader family of nearly optimal codebooks with tunable parameters (Heng, 2017).
• Trace Functions and Field Extensions: The trace map projects elements from larger fields to subfields, allowing subfield trace indicators to parametrize codewords with desirable correlation. Such constructions allow flexible lengths and manageable alphabet sizes (Wu et al., 2019).
• Ring-based Characters: Extension of character theory to local rings (e.g., $R = \mathbb{F}_q + u\mathbb{F}_q$) generates new codebook families with parameter settings unattainable in strict field constructions (Qian et al., 2019).

These frameworks allow both flexibility (in choosing code length, codebook size, and alphabet) and explicit control over the mathematical objects governing correlation properties.

2. Cross-Correlation Bounds and Asymptotic Optimality

A central metric in FC Complexity is the maximum cross-correlation amplitude, $I_{\max}$, between distinct codewords. This directly quantifies worst-case interference in multi-user systems and the coherence in compressed sensing. The Welch bound,

$I_w = \sqrt{\frac{N-K}{K(N-1)}},$

serves as a theoretical lower bound (for $(N, K)$ codebooks). The Levenshtein bound provides a tighter constraint in some parameter regimes.

Major results include:

• Nearly Optimal Codebooks: By deriving explicit formulas for inner products—often using the properties of Gauss or hybrid character sums—constructions achieve

$\lim_{q \to \infty} \frac{I_{\max}}{I_w} = 1,$

showing asymptotic optimality (Heng, 2017, Lu et al., 2019, Heng et al., 29 Jun 2025).

• Two/Three-Valued Correlation: Some constructions lead to codebooks where inner products take only two or three values. This rigid structure is highly favorable for predictability in interference and for engineering equiangular tight frames (Heng et al., 29 Jun 2025).
• Parameter Flexibility: By varying field sizes $q$, extension degrees $m_i$, and code construction parameters, one can optimize trade-offs among codebook length, codeword count, and cross-correlation (Heng, 2017, Wu et al., 2019).

3. Computational Complexity and Algorithmic Barriers

Recent advances have formalized the computational complexity of constructing or even describing sequences of codes or rates achieving near-optimal performance in finite codebook or blocklength regimes. Complexity classes tailored to unary encoding—FP$_1$ (polynomial time function computable) and $\#$P$_1$ (counting class)—become relevant.

Key findings:

• Sequence Hardness: While a single codebook or achievable rate at fixed parameters might be polynomial-time computable, the sequence $\{R_{n_M}(\epsilon)\}_{n_M}$ attaining distances to capacity of $1/2^M$ or the corresponding $\{n_M\}$ sequence is not polynomial-time computable under the widely believed assumption FP$_1$ $\neq$ $\#$P$_1$ (Boche et al., 10 Jul 2024). The computation becomes super-polynomial in parameter $M$.
• Capacity Approximations: For additive Gaussian noise channels with computable noise/power parameters, either the achievable rate or blocklength sequence converging to capacity exhibits complexity exceeding any polynomial bound in the desired precision.
• Implications: This theoretical boundary compels practical applications to accept only approximate capacity-achieving codes, as exact constructions for arbitrary precision become infeasible to enumerate or optimize.

4. Codebooks from Bent and Dual-Bent Functions

Vectorial dual-bent functions have prominent utility due to their extremal spectral properties. Their application in hybrid character sum constructions permits the explicit calculation of cross-correlation amplitude and grants precise control over codebook performance:

• Hybrid Character Sums: Sums of the form

$\sum_{x \in V_n^{(p)}} \psi(F(x))\chi_{1}(a x)$

(with $F$ vectorial dual-bent and $\psi$ a multiplicative character) can be made take only $0$ or $\pm p^{n/2}$, yielding codebooks with just two- or three-valued amplitude structure (Heng et al., 29 Jun 2025).

• Small Alphabet Size: Such constructions enable asymptotically optimal codebooks even for small $p$ (e.g., $p=3$), minimizing the hardware or storage complexity for implementations.
• Applications: Their predictable amplitude profile makes them valuable for applications in unitary space–time coding, spread-spectrum communications, compressed sensing, and quantum information.

5. Covering Codes, Coefficient Complexity, and Distributed Computation

Expanding the notion of "codebook" beyond signal sets for communications, recent work frames finite codebook complexity in distributed computation contexts using covering codes:

• Non-Binary Covering Codes: Covering codes over $\mathbb{F}_p$ with small covering radius enable storage schemes that can reconstruct arbitrary linear functions (with coefficients in a fixed finite set $A$) by accessing few storage nodes (Ramkumar et al., 9 May 2024).
• Coefficient Complexity: The $p$-complexity $C_p(A)$ quantifies the minimal "additive decomposition" depth of coefficient sets $A$ using $p$-term arithmetic progressions, directly scaling the "access" cost in universal distributed schemes.
• Access-Redundancy Tradeoff: By combining codeword reduction (identifying equivalent classes under negation, etc.) and amalgamation of base covering codes, constructions yield better redundancy-access tradeoffs and new feasible points in the design space—such as accessing only $\beta = 1/2$ fraction of storage nodes for ternary computations, with redundancy $\alpha = 2$, outperforming naive schemes.

Such advances highlight the generalized role of finite codebooks in orchestrating efficient distributed computations under alphabet and coefficient constraints.

6. Practical and Theoretical Significance

Advances in finite codebook complexity have deep implications across communications theory, signal processing, distributed computation, and quantum information:

• Wireless Systems: SCMA codebooks using sparse, structured lattice constellations achieve substantial shaping and interference-canceling gains—even for moderate-size finite sets, with decoding complexity scaling as $M^{d_f}$, where $d_f$ is the local codeword density (Taherzadeh et al., 2014).
• Compressed Sensing & Frame Theory: Nearly equiangular tight frames and deterministic sensing matrices constructed from codebooks with near-Welch-bound cross-correlation boost sparse recovery guarantees and measurement efficiency.
• Quantum Information: Construction of POVMs and SIC structures (where two- or three-valued cross-correlation is desirable) benefits from the same framework as wireless and code-division systems.
• Algorithmic Caution: As the computational hardness of finely optimizing codebook sequences is now precisely formalized, practical system design strategies must emphasize tractable approximations, parameter tuning, and exploitation of universal protocols to sidestep intractable growth in construction, storage, or decoding complexity.

7. Outlook and Future Directions

Ongoing research continues to generalize FC complexity, exploring new algebraic frameworks (e.g., beyond finite fields and local rings), tighter integration of computational complexity theory for practical constraints, and expanding the repertoire of nearly-optimal constructions with novel parameter sets. Multidisciplinary approaches—combining additive combinatorics (as in $p$-complexity), algebraic coding, probabilistic method, and computer science complexity theory—offer promising avenues to manage the inherent tensions among optimality, complexity, and practicality in codebook design for future high-throughput, robust, and resource-limited systems.