Leurent and Sibleyras reduced low-data key recovery on the two-round Even–Mansour construction to a structured 3-XOR problem. In the very-low-data regime, their attack uses D = λn known plaintext/ciphertext pairs, T = Q = Θ(2n/(λn)) time and public-permutation queries, and a Θ(2λn)-sized table. We study the fixed-rank matching step in that reduction. After the rank-r coordinate transformation, each useful prefix induces a shifted-claw problem on the r-dimensional hypercube. Thus the full 2r suffix table can be replaced, at fixed rank, by a translated block A ⊆ Fr>2 and the closed unit neighborhood N(A). For arbitrary nonempty A, the randomized translated-block matcher uses M = O(|A|) + poly(n), Q = O (2n/μA (1 + |N (A)/|A| )︃)︃ polylog(n), where μA is an exact multiplicity parameter satisfying μ ≤ μA ≤ r. We derive the expectation of μA in the independent-endpoint model and an exact interpolation formula for the simplicial memory profile between consecutive Hamming-ball volumes. We also prove that, in the same model, when r|A|/2n = o(1), the exact multiplicity parameter satisfies μA = (1 − o(1))r with probability 1 − o(1). Hamming balls give closed-form memory profiles, Harper’s vertex-isoperimetric theorem gives minimum conservative overheads at fixed block size within the closed-neighborhood translated-block model, and translational covers give deterministic fixed-prefix variants. A coding-theoretic interpretation identifies translated Hamming-ball slices with covering-code decoding regions and linear-code standard arrays with deterministic syndrome-leader slices. The results refine the memory profile of the fixed-rank matcher; they do not improve the data-query trade-off of the underlying low-data attack.
Let d(n, k, q) (dSO(n, k, q)) be the highest minimum weight of linear (self-orthogonal) [n, k]q codes. Then it is interesting to know if d(n, k, q) = dSO(n, k, q). Since the pioneering work of V. Pless in 1972, the classification of optimal binary self-orthogonal codes has been studied in many papers. There were some partial results on the problem dSO(n, k, 2) = d(n, k, 2) for binary self-orthogonal codes of low dimensions k ≤ 6. Some self-orthogonal q-ary codes were constructed in a series of recent papers published in IEEE Trans. Inf. Theory and a few of them are optimal codes. In this paper, we construct many self-orthogonal cyclic q-ary codes with optimal, best known, almost optimal or almost best known minimum weights. Infinite families of self-orthogonal cyclic q-ary codes with rates around 1/4 and the minimum weights at least n/2logqn are also constructed. From constructed selforthogonal ternary cyclic codes, we also construct many ternary LCD codes with optimal, almost optimal or suboptimal minimum weights. Applications to quantum stabilizer codes and t-designs are also discussed.
In decentralized network control, communication plays a crucial role by transforming local observations into shared knowledge, enabling agents to coordinate their actions. This paper investigates how communication facilitates cooperation behavior and therefore improves the overall performance inWitsenhausen counterexample, a canonical toy problem in distributed decision-making. We consider the vector-valued setups where the encoder, i.e., the first decision-maker (DM) acts causally and the decoder, i.e., the second DM, operates noncausally, 1) without and 2) with access to channel feedback. Using a coordination coding framework, we characterize the achievable power-estimation cost regions in single-letter expressions for both scenarios. We show that feedback enlarges the coordination region compared with the no-feedback setup. Building on the characterization of the power-estimation cost region, we propose a hybrid strategy that combines discrete quantization with a continuous Gaussian codebook, the Zero Estimation Cost (ZEC) strategy. This strategy uses coding tools that allow perfect reconstruction of the target symbols, leading to an asymptotic estimation cost equal to zero, while substantially reducing the power cost. Moreover, when channel feedback is available, we propose an analogous ZEC strategy that simultaneously achieves zero power and zero estimation costs in the low-noise regime. Finally, we establish that, when restricted to jointly Gaussian random variables, the performance is identical across all setups featuring at least one causal DM, regardless of the presence of feedback information.
Let Xn be uniform on the binary cube and let Yn be obtained from Xn through a memoryless binary symmetric channel with crossover probability α. We study the fixed-k quantization problem Mk(n, α) = maxf:{0,1}n→{0,1}k I(f(Xn); Yn) , which was introduced by Chandar and Tchamkerten and interpolates between the one-bit Courtade–Kumar problem and the (n − 1)-bit endpoint of Huleihel and Ordentlich. We first record a concise affine benchmark and show that nontrivial modulo-2 mixing cannot improve on coordinate transmission. For arbitrary quantizers, we derive the exact second- and fourth-order high-noise coefficients. The leading coefficient Γ(f) = Σu pu Σn i=1 E[(−1)Xi | f(Xn) = u]2 is the weighted aggregate of the level-1 Fourier masses of the partition cells, while the quartic coefficient Θ(f) separates centered level-2 dependence from marginal saturation. We substantially extend the structural analysis by introducing a Φ-mutual-information tensorization framework. It yields a variational family of upper bounds on Γ(f),every two-bit quantizer> including the Shannon bound, the support-size bound, and an explicit power-divergence interpolation. Using the equivalence between level-1 Fourier weight and minimum average distance, we further obtain profile-sensitive bounds from the results of Fu, Wei, and Yeung and Lei Yu. To the best of our knowledge, the Φ-tensorization framework and the average-distance connection are new in this multibit setting; together they settle the second-order coefficient completely for k = 2: every two-bit quantizer satisfies Γ(f) ≤ 2, with equality attained by equiprobable four-cell subcube partitions. We also prove the exact identity I ( f(Xn); Y n) = ( 1 − h2(α) ) H ( f(Xn) ) for subcube quantizers, meaning quantizers whose nonempty cells are subcubes. Hence nonlinear quantizers can attain the projection benchmark, and the fixed-k question is a value problem rather than a uniqueness problem. Computational evidence and quartic obstructions further reduce the unresolved regime to large-support, genuinely non-subcube partitions with nonzero conditional pairwise dependence.
An asynchronous K-active-user unsourced multiple access channel constitutes a fundamental model for uncoordinated massive access in future wireless networks. We consider scenarios in which the first symbol of each transmitted codeword arrives at the receiver no later than the (1 + D)-th channel use, and investigate both latency-constrained and latency-unconstrained decoding regimes. In the latency-constrained setting, all transmitted codewords must be decoded within the first n channel uses, whereas in the latency-unconstrained setting, decoding is performed after observing n + D channel uses. Furthermore, we investigate the impact of unknown delay information at the receiver. The joint effect of asynchronicity and unknown delays requires the analysis of all codeword-delay combinations, leading to an analysis whose complexity scales exponentially with K and polynomially with D. To address this challenge, we derive a uniform upper bound on the per-user probability of error that applies to all codeword-delay combinations and significantly simplifies the analysis. In addition, we characterize the trade-off between the required energy per bit and the number of active users under various delay and latency constraints. For the considered numerical setup, our achievability bounds indicate that latency-unconstrained decoding reduces the required Eb/N0 by approximately 2.8 dB compared with latency-constrained decoding for K = 300. Moreover, under wrap decoding, the lack of delay information incurs an additional loss of approximately 0.4 dB relative to the case of perfect delay knowledge. The results further suggest that asynchronicity is associated with an increase in the required Eb/N0 and shifts the threshold number of active users—below which the required Eb/N0 remains nearly constant—toward smaller values.
