SOCC: Codebook Design for XL-RIS MIMO Beam Training

• The paper demonstrates that SOCC decomposes joint optimization into separate stages for BS precoding and RIS phase shifts, balancing tractability with optimal performance.
• The method fixes the BS precoder based on the BS-RIS channel before optimizing the RIS phases to generate a hierarchical multi-resolution codebook for beam training.
• SOCC reduces design complexity and beam-training overhead, with parallels in distortion-aware ECC and group-wise vector quantization despite inherent trade-offs versus joint optimization.

Separately Optimized Codebook Construction (SOCC) denotes a codebook-design methodology in which coupled design variables are decomposed into sequential or blockwise optimization stages rather than being optimized in a single fully joint program. The term is introduced explicitly for discrete XL-RIS-aided near-field MIMO beam training, where base-station (BS) precoding and RIS phase-shift codewords are designed separately within a multi-resolution codebook framework (Zhang et al., 26 Aug 2025). In a broader methodological sense, closely related architectures also appear in distortion-aware error-correcting code design, group-wise vector-quantized codebooks, and algebraic codebook families, even when the term SOCC is not used. Across these settings, the common pattern is to fix one part of the system by an analytically or structurally motivated rule, then optimize the remaining codebook variables under that fixed choice; the main technical issue is the resulting tradeoff between tractability and the loss of full joint optimality (Wu, 2018, Zheng et al., 15 Oct 2025, Wei et al., 29 Jun 2026).

1. Definition and conceptual scope

In its explicit form, SOCC is a codebook-construction method for a discrete XL-RIS-aided near-field MIMO downlink system with a BS having $M$ antennas, an XL-RIS with $N=N_2N_1$ reflecting elements, and $K$ single-antenna users, under blocked direct BS-user links and discrete RIS phases

$$\theta_n \in \left\{0,\frac{2\pi}{2^v},\dots,\frac{(2^v-1)2\pi}{2^v}\right\},\qquad \phi_n=e^{j\theta_n},\quad n=1,\dots,N.$$

The central codebook problem is to construct a hierarchical near-field multi-resolution codebook that supports coarse-to-fine beam training while accounting for both BS precoding and RIS phase shifts under discrete hardware constraints (Zhang et al., 26 Aug 2025).

Within that paper’s terminology, SOCC is defined relative to JOCC. JOCC jointly optimizes the BS precoder and RIS phase shifts for each codeword, whereas SOCC first fixes a BS precoder by aligning it with the BS-RIS channel and then optimizes the RIS phase shifts for the desired beam pattern. The paper states that JOCC yields “the most superior beam training performance,” while SOCC achieves higher performance than a single-antenna BS codebook “at a similar complexity” (Zhang et al., 26 Aug 2025).

Methodologically, the same decomposition appears in related but terminologically different settings. In distortion-aware error correction, a fixed codebook induces a Bayes-optimal decoder under a task loss, and codebooks are then searched in an outer loop using a distortion-aware surrogate objective (Wu, 2018). In Group-VQ, a codebook is partitioned into sub-codebooks that are independently updated across groups but jointly parameterized within groups, yielding a block-granular analogue of separate optimization (Zheng et al., 15 Oct 2025). By contrast, the semantic-communication codebook in (Wang et al., 8 Oct 2025) is trained with a single end-to-end joint objective, which clarifies what SOCC is not: it is not full joint optimization over all codebook-related criteria.

This broader usage suggests an Editor’s term, “SOCC-like decomposition,” for designs that separate optimization across subsystems, groups, or stages without necessarily making each codeword individually independent. A plausible implication is that SOCC is best understood as a family of design decompositions rather than a single algorithmic template.

2. Formal SOCC in discrete XL-RIS near-field MIMO

The SOCC formulation in (Zhang et al., 26 Aug 2025) is built on the beam-training signal model

$\bm x_t = \bm w_t s,$

with received user signal

$$y_u = \bm\phi^{\rm H} \bm H_u(x_u,y_u,z_u)\bm w_t s + n_u,$$

where

$$\bm H_u(x_u,y_u,z_u)=\operatorname{diag}\!\big(\bm h_u(x_u,y_u,z_u)\big)^{\rm H}\bm G \in \mathbb C^{N\times M}.$$

The beam-training metric is the received beam gain

$$\left|\bm\phi^{\rm H}\bm H_u(x_i,y_u,z_k)\bm w_t\right|^2/\sigma_u^2.$$

Because the regime is near-field, the codebook must shape energy over spatial regions in the $(x,z)$-plane rather than over angular sectors only (Zhang et al., 26 Aug 2025).

For hierarchical beam training, level-$l$ sampling points are

$N=N_2N_1$0

with target coverage regions

$N=N_2N_1$1

The desired beam pattern is represented by amplitude vector $N=N_2N_1$2 and auxiliary phase vector $N=N_2N_1$3, where

$N=N_2N_1$4

Thus a codeword is intended to produce approximately constant high gain inside a target spatial cell and suppress gain outside it (Zhang et al., 26 Aug 2025).

SOCC then decomposes codebook construction into two steps. First, it optimizes the BS precoder independently through

$N=N_2N_1$5

whose optimal solution is

$N=N_2N_1$6

Second, with $N=N_2N_1$7 fixed, it optimizes the RIS phase vector and desired beam phases by solving

$N=N_2N_1$8

where

$N=N_2N_1$9

This is the defining separation: the BS-side beam is fixed once by the BS-RIS channel, and only the RIS-side codeword is specialized to the target region (Zhang et al., 26 Aug 2025).

The paper also gives a unified formulation in which JOCC optimizes $K$0, whereas SOCC optimizes $K$1 with $K$2 fixed, yielding the inequality

$K$3

Accordingly, SOCC is explicitly a decomposition-based approximation to JOCC rather than an alternative optimum. The paper’s Remark 1 states that SOCC performance depends on the singular values of $K$4: the more dominant the largest singular values are, the closer SOCC is to JOCC (Zhang et al., 26 Aug 2025).

3. Algorithmic workflow, solver structure, and hierarchical deployment

The SOCC workflow is embedded in a hierarchical beam-training pipeline. For each search layer, one first specifies the sampling region, grid sizes, target coverage $K$5, and desired gain level $K$6. One then computes the common BS codeword $K$7, builds sampled cascaded responses $K$8, and solves the RIS-only pattern-matching problem for each spatial cell. The resulting layer-wise codewords are stored for coarse-to-fine search during online training (Zhang et al., 26 Aug 2025).

Although the paper does not provide a standalone pseudocode box titled specifically for SOCC, it states that the SOCC problem can be solved by the same framework developed for JOCC. In that framework, RIS-phase optimization uses IPDD with auxiliary variable $K$9, via the iterations

$$\theta_n \in \left\{0,\frac{2\pi}{2^v},\dots,\frac{(2^v-1)2\pi}{2^v}\right\},\qquad \phi_n=e^{j\theta_n},\quad n=1,\dots,N.$$0

$$\theta_n \in \left\{0,\frac{2\pi}{2^v},\dots,\frac{(2^v-1)2\pi}{2^v}\right\},\qquad \phi_n=e^{j\theta_n},\quad n=1,\dots,N.$$1

$$\theta_n \in \left\{0,\frac{2\pi}{2^v},\dots,\frac{(2^v-1)2\pi}{2^v}\right\},\qquad \phi_n=e^{j\theta_n},\quad n=1,\dots,N.$$2

with closed-form quadratic update

$$\theta_n \in \left\{0,\frac{2\pi}{2^v},\dots,\frac{(2^v-1)2\pi}{2^v}\right\},\qquad \phi_n=e^{j\theta_n},\quad n=1,\dots,N.$$3

Discrete phase projection uses the CMDPP rule, and the auxiliary beam phase is updated by phase alignment,

$$\theta_n \in \left\{0,\frac{2\pi}{2^v},\dots,\frac{(2^v-1)2\pi}{2^v}\right\},\qquad \phi_n=e^{j\theta_n},\quad n=1,\dots,N.$$4

In SOCC, the JOCC update for $$\theta_n \in \left\{0,\frac{2\pi}{2^v},\dots,\frac{(2^v-1)2\pi}{2^v}\right\},\qquad \phi_n=e^{j\theta_n},\quad n=1,\dots,N.$$5 is simply removed because $$\theta_n \in \left\{0,\frac{2\pi}{2^v},\dots,\frac{(2^v-1)2\pi}{2^v}\right\},\qquad \phi_n=e^{j\theta_n},\quad n=1,\dots,N.$$6 is fixed (Zhang et al., 26 Aug 2025).

The deployment logic is hierarchical rather than exhaustive. If $$\theta_n \in \left\{0,\frac{2\pi}{2^v},\dots,\frac{(2^v-1)2\pi}{2^v}\right\},\qquad \phi_n=e^{j\theta_n},\quad n=1,\dots,N.$$7 denotes the number of search levels and $$\theta_n \in \left\{0,\frac{2\pi}{2^v},\dots,\frac{(2^v-1)2\pi}{2^v}\right\},\qquad \phi_n=e^{j\theta_n},\quad n=1,\dots,N.$$8 the number of searched regions per level, the overhead is $$\theta_n \in \left\{0,\frac{2\pi}{2^v},\dots,\frac{(2^v-1)2\pi}{2^v}\right\},\qquad \phi_n=e^{j\theta_n},\quad n=1,\dots,N.$$9, compared with $\bm x_t = \bm w_t s,$0 for exhaustive search. The paper states that hierarchical training reduces overhead by roughly $\bm x_t = \bm w_t s,$1 (Zhang et al., 26 Aug 2025). This matters because SOCC is not a search procedure itself; it is the codebook-construction rule that supplies the beam patterns used by the hierarchical search.

Empirically, the paper reports that both JOCC and SOCC can obtain beam patterns that closely match the desired multi-resolution pattern. It states that “with the proposed algorithm, we can obtain an exact match of the desired beam pattern by either the JOCC method or the SOCC method.” It further reports that SOCC “can greatly reduce the design complexity compared to the proposed JOCC method,” that its complexity is close to the SA-BS codebook baseline, and that its beam-training performance is “vastly improved” over the SA-BS codebook (Zhang et al., 26 Aug 2025).

4. SOCC-like decomposition in distortion-aware error-correcting code design

A strong methodological precursor to SOCC appears in distortion-aware ECC design, where the message alphabet is a finite set

$\bm x_t = \bm w_t s,$2

a codebook is a bijection

$\bm x_t = \bm w_t s,$3

and decoding returns

$\bm x_t = \bm w_t s,$4

The key departure from standard BER-centric design is the source-space distortion

$\bm x_t = \bm w_t s,$5

with example losses

$\bm x_t = \bm w_t s,$6

The paper’s central claim is that when bit positions have unequal semantic significance, Hamming-distance-based geometry is misaligned with the communication objective (Wu, 2018).

For a fixed codebook $\bm x_t = \bm w_t s,$7, the decoder is selected from a Bayesian criterion: $\bm x_t = \bm w_t s,$8 By setting

$\bm x_t = \bm w_t s,$9

Bayes decoding becomes the principled decoder for minimizing $$y_u = \bm\phi^{\rm H} \bm H_u(x_u,y_u,z_u)\bm w_t s + n_u,$$0. For integer-difference losses, the Bayes estimator reduces to the posterior mean for squared loss and the posterior median for absolute loss (Wu, 2018).

Codebook optimization then occurs in an outer loop using the surrogate

$$y_u = \bm\phi^{\rm H} \bm H_u(x_u,y_u,z_u)\bm w_t s + n_u,$$1

and for AWGN this is approximated by

$$y_u = \bm\phi^{\rm H} \bm H_u(x_u,y_u,z_u)\bm w_t s + n_u,$$2

The search is heuristic: “several search algorithms including genetic algorithm and various types of hill-climbing algorithms” were tested, with a simple genetic algorithm reported as best; candidate codebooks are represented either directly as concatenated codewords or, in the linear setting, by a generator matrix $$y_u = \bm\phi^{\rm H} \bm H_u(x_u,y_u,z_u)\bm w_t s + n_u,$$3 (Wu, 2018).

This architecture is not labeled SOCC in the paper, but it is very close to a decoder-for-fixed-codebook plus outer-loop codebook-search decomposition. The decoder is not independent of the codebook, because $$y_u = \bm\phi^{\rm H} \bm H_u(x_u,y_u,z_u)\bm w_t s + n_u,$$4 depends on $$y_u = \bm\phi^{\rm H} \bm H_u(x_u,y_u,z_u)\bm w_t s + n_u,$$5. However, once $$y_u = \bm\phi^{\rm H} \bm H_u(x_u,y_u,z_u)\bm w_t s + n_u,$$6 is fixed, decoding is analytically specified, and the outer search optimizes the codebook with respect to a distortion-aware surrogate rather than a joint end-to-end parameterization. The paper’s treatment of unrestricted versus linear codebooks further sharpens the SOCC lesson: unrestricted mappings are more expressive, but generator-matrix search can converge faster and even outperform unrestricted search under a fixed optimization budget (Wu, 2018). This suggests that structural restrictions may improve practical separately optimized codebook construction when search cost dominates asymptotic expressiveness.

5. Blockwise SOCC analogues in vector-quantized models and contrasts with joint codebook learning

In vector-quantized models, the closest analogue to SOCC is Group-VQ, which partitions the codebook

$$y_u = \bm\phi^{\rm H} \bm H_u(x_u,y_u,z_u)\bm w_t s + n_u,$$7

and treats each group as “the smallest unit independently updated during training.” For a code $$y_u = \bm\phi^{\rm H} \bm H_u(x_u,y_u,z_u)\bm w_t s + n_u,$$8, the paper states that for $$y_u = \bm\phi^{\rm H} \bm H_u(x_u,y_u,z_u)\bm w_t s + n_u,$$9,

$$\bm H_u(x_u,y_u,z_u)=\operatorname{diag}\!\big(\bm h_u(x_u,y_u,z_u)\big)^{\rm H}\bm G \in \mathbb C^{N\times M}.$$0

so that

$$\bm H_u(x_u,y_u,z_u)=\operatorname{diag}\!\big(\bm h_u(x_u,y_u,z_u)\big)^{\rm H}\bm G \in \mathbb C^{N\times M}.$$1

Within each group, however, optimization remains joint through

$$\bm H_u(x_u,y_u,z_u)=\operatorname{diag}\!\big(\bm h_u(x_u,y_u,z_u)\big)^{\rm H}\bm G \in \mathbb C^{N\times M}.$$2

with group-specific core $$\bm H_u(x_u,y_u,z_u)=\operatorname{diag}\!\big(\bm h_u(x_u,y_u,z_u)\big)^{\rm H}\bm G \in \mathbb C^{N\times M}.$$3, projector $$\bm H_u(x_u,y_u,z_u)=\operatorname{diag}\!\big(\bm h_u(x_u,y_u,z_u)\big)^{\rm H}\bm G \in \mathbb C^{N\times M}.$$4, and bias $$\bm H_u(x_u,y_u,z_u)=\operatorname{diag}\!\big(\bm h_u(x_u,y_u,z_u)\big)^{\rm H}\bm G \in \mathbb C^{N\times M}.$$5 (Zheng et al., 15 Oct 2025).

This yields a continuum between full separate and full joint optimization. Vanilla VQ corresponds to

$$\bm H_u(x_u,y_u,z_u)=\operatorname{diag}\!\big(\bm h_u(x_u,y_u,z_u)\big)^{\rm H}\bm G \in \mathbb C^{N\times M}.$$6

while Joint VQ corresponds to

$$\bm H_u(x_u,y_u,z_u)=\operatorname{diag}\!\big(\bm h_u(x_u,y_u,z_u)\big)^{\rm H}\bm G \in \mathbb C^{N\times M}.$$7

Group-VQ occupies the intermediate regime $$\bm H_u(x_u,y_u,z_u)=\operatorname{diag}\!\big(\bm h_u(x_u,y_u,z_u)\big)^{\rm H}\bm G \in \mathbb C^{N\times M}.$$8. Quantization remains globally competitive over the full codebook rather than using explicit routing, so the method is best described as independent across groups, joint within each group (Zheng et al., 15 Oct 2025).

The paper’s empirical results support the claim that moderate separation can improve the utilization–reconstruction tradeoff. On ImageNet-1k with a VQGAN backbone and codebook size $$\bm H_u(x_u,y_u,z_u)=\operatorname{diag}\!\big(\bm h_u(x_u,y_u,z_u)\big)^{\rm H}\bm G \in \mathbb C^{N\times M}.$$9, SimVQ with group $$\left|\bm\phi^{\rm H}\bm H_u(x_i,y_u,z_k)\bm w_t\right|^2/\sigma_u^2.$$0 achieved rFID $$\left|\bm\phi^{\rm H}\bm H_u(x_i,y_u,z_k)\bm w_t\right|^2/\sigma_u^2.$$1, LPIPS $$\left|\bm\phi^{\rm H}\bm H_u(x_i,y_u,z_k)\bm w_t\right|^2/\sigma_u^2.$$2, PSNR $$\left|\bm\phi^{\rm H}\bm H_u(x_i,y_u,z_k)\bm w_t\right|^2/\sigma_u^2.$$3, SSIM $$\left|\bm\phi^{\rm H}\bm H_u(x_i,y_u,z_k)\bm w_t\right|^2/\sigma_u^2.$$4, while Group-VQ with group $$\left|\bm\phi^{\rm H}\bm H_u(x_i,y_u,z_k)\bm w_t\right|^2/\sigma_u^2.$$5 achieved rFID $$\left|\bm\phi^{\rm H}\bm H_u(x_i,y_u,z_k)\bm w_t\right|^2/\sigma_u^2.$$6, LPIPS $$\left|\bm\phi^{\rm H}\bm H_u(x_i,y_u,z_k)\bm w_t\right|^2/\sigma_u^2.$$7, PSNR $$\left|\bm\phi^{\rm H}\bm H_u(x_i,y_u,z_k)\bm w_t\right|^2/\sigma_u^2.$$8, SSIM $$\left|\bm\phi^{\rm H}\bm H_u(x_i,y_u,z_k)\bm w_t\right|^2/\sigma_u^2.$$9. In an ablation with codebook size $(x,z)$0, the paper reports that group size $(x,z)$1 is optimal in that setup; too many groups make Group-VQ degenerate toward Vanilla VQ and reintroduce codebook collapse (Zheng et al., 15 Oct 2025).

By contrast, the semantic-communication codebook in (Wang et al., 8 Oct 2025) provides a useful counterexample. There, the codebook

$(x,z)$2

is trained under a single codebook loss

$(x,z)$3

combining quantization fidelity, channel-aware semantic distortion, and entropy regularization. The full model then jointly optimizes $(x,z)$4 end to end (Wang et al., 8 Oct 2025). This contrast is instructive: although the paper decomposes the design criteria conceptually, it does not separate optimization stages. A plausible implication is that SOCC becomes harder to justify when Voronoi partitions, occupancy statistics, and channel robustness all co-adapt through the same codebook geometry.

6. Algebraic modularity, design tradeoffs, and limitations

A different perspective on SOCC comes from deterministic algebraic codebook families constructed from vectorial dual-bent functions. The paper studies codebooks

$(x,z)$5

with objective

$(x,z)$6

benchmarking against the Welch bound

$(x,z)$7

Its constructions use modular ingredients: a vectorial dual-bent function $(x,z)$8, support sets such as $(x,z)$9, additive and multiplicative characters, subset choices $l$0, permutations $l$1, and product or lifted constructions involving one or two base functions (Wei et al., 29 Jun 2026).

This work does not use the SOCC terminology, but it is strongly modular. For example, in the additive-character support construction,

$l$2

the support $l$3 can be selected separately through a set $l$4 in

$l$5

while in mixed-character constructions a permutation $l$6 independently controls how multiplicative labels are assigned over $l$7 and $l$8 partitions (Wei et al., 29 Jun 2026). The resulting inner products remain analyzable because the character sums collapse to formulas determined by $l$9.

This suggests a broader interpretation of SOCC: separate optimization need not be numerical or iterative; it can also be algebraic and compositional, with independently chosen primitives whose interactions remain exactly tractable. In that sense, (Wei et al., 29 Jun 2026) provides a correlation-controlled modular template, while (Zhang et al., 26 Aug 2025) provides an explicit systems-level SOCC instance, (Wu, 2018) provides a decoder-fixed outer-loop search paradigm, and (Zheng et al., 15 Oct 2025) provides a blockwise compromise between full coupling and full separation.

Across these formulations, the principal limitation is consistent. Full joint optimization can only improve, or at worst match, the objective attained by a constrained separate scheme; in the XL-RIS paper this is formalized by

$N=N_2N_1$00

(Zhang et al., 26 Aug 2025). Separate optimization is therefore justified primarily by tractability, storage, convergence, or hardware constraints rather than by universal optimality. The data also indicate three recurring failure modes: excessive separation can reduce utilization or reintroduce collapse in vector-quantized models (Zheng et al., 15 Oct 2025); finite search budgets can make unrestricted outer-loop codebook search inferior to more structured families in ECC (Wu, 2018); and SOCC quality in XL-RIS beam training depends on whether the BS-RIS channel $N=N_2N_1$01 is dominated by its strongest singular mode (Zhang et al., 26 Aug 2025).

Taken together, these results define SOCC as a technically specific but conceptually broad design principle: specify the subsystem or block to be fixed, derive the optimal or structured rule for that block, and optimize the remaining codebook variables under the resulting reduced problem. The main scientific question is not whether separation exists, but at what granularity it remains accurate enough to preserve the task metric while materially reducing complexity.