communication system engineers

Abstract: We study the maximal length of MDS algebraic geometry codes from genus-two curves over finite fields. Let $C_\mathcal{ L}(D,G)$ be such a code over $\mathbb F_q$, with length $n$ and dimension $k$. Assume that $3\leq k\leq n-3$. We prove that $n<(2q+10\sqrt q+16)/3$. This bound holds for every prime power $q$ and for both parities of $k$. It gives a uniform upper bound of $2q/3+O(\sqrt q)$ for the maximal length. The explicit bound is strictly smaller than $q+1$ when $q\geq125$.

Abstract: We establish polarization for memoryless sources taking values on the unit-perimeter circle, or on more general compact groups, under Arıkan's polar transform. For sources with finite differential entropy, all but an $O((\log N)^{-1/12})$ fraction of the conditional differential entropies in successive cancellation order lie within any fixed positive distance of zero, where $N$ is the block length. Examples indicate that the true polarization rate is indeed very slow: an inverse-logarithmic rate does not underestimate the polarization speed at moderate block lengths.

Abstract: Soft information perturbations can significantly improve the successive cancellation (SC) decoding performance of polar codes. This is an effective method for realizing low-latency yet high performance for long polar codes, since each perturbation-based SC (PSC) decoding can be performed in parallel. However, little theoretical insight has been revealed so far regarding its work rationale. This paper provides the theoretical analysis of the soft information perturbation. It reveals that PSC decoding improves the SC decoding performance through correcting the erroneous bit estimations, especially the first erroneous bit (FEB). It further validates that perturbing the soft information does not degrade the error-correction capability of SC decoding. Such revelations lead to the design of optimal perturbations that maximizes the FEB correction probability, resulting in an enhanced PSC decoding performance. Based on this, a novel hybrid PSC (HPSC) decoding is proposed. By introducing the random and biased perturbations, the HPSC decoding can better divert from the initial erroneous estimation and converge to the intended one. Our simulation results demonstrate that with the optimal perturbation, the HPSC decoding can achieve a similar performance as the cyclic redundancy check (CRC) aided SC list (CA-SCL) decoding, while avoiding path sorting and expansion.

Abstract: We develop information--estimation relations for an arbitrary discrete memoryless channel with finite input and output alphabets by varying the expected number of observations while keeping the channel fixed. A single hidden input is observed through a Poisson number of conditionally independent channel uses, whose mean specifies the observation budget. The derivative of mutual information with respect to this budget equals the minimum expected relative-entropy loss for estimating the input's output distribution. The posterior average of the channel's output distributions achieves this minimum and is the Bayesian predictor for the next observation. When the input distribution, the channel, or both are misspecified, the relative entropy between the true and assumed observation laws is the integral of the excess prediction loss over the budget. We also extend these relations to finite-state processes taking values in the input alphabet and observed through the channel at independent Poisson sampling times. Both in the matched setting and under model mismatch, the expected prediction loss accumulated over time using past observations equals the corresponding loss using the full observation record (noncausal estimation), averaged over lower sampling rates.

Abstract: We propose a novel soft-aided hard-decision decoding scheme for product codes based on two-bit quantized reliability markers. In particular, one bit is used for a two-level tolerance miscorrection check to validate the bounded distance decoding (BDD) result, while the other identifies bits to be randomly flipped upon BDD failure and miscorrection. List decoding is incorporated to further facilitate the error performance. Density evolution is performed for the design of the quantization thresholds. Compared to iterative BDD, the proposed scheme requires only moderate increases in memory and computational overhead. Simulation results verify that our scheme outperforms several unquantized soft-aided decoders, while maintaining small memory costs and low BDD attempts.

Abstract: We propose a novel low-complexity soft-aided hard-decision decoding scheme for product-like codes, termed iterative bounded-distance decoding (iBDD) with graded scaled reliability and bit-flipping (GSRF). This scheme operates on four-level quantized soft information. It incorporates a graded scaled reliability rule to assign decoding confidence to the bounded-distance decoding result, and employs a bit-flipping rule based on bit-level and component-code-level reliability. A selective re-decoding rule is designed to further facilitate the error performance while keeping the number of BDD attempts in reasonably low levels. The decoding parameter design of iBDD-GSRF is investigated for product codes and staircase codes, respectively. Simulation results on both codes verify that iBDD-GSRF achieves a gain of more than 0.4 dB over the conventional iBDD, comparable to several existing unquantized soft-aided decoders.

Abstract: This paper investigates a shift-sum decoding algorithm that employs dual codewords. Existing dual codeword shift-sum decoding schemes mainly rely on minimum-weight dual codewords. However, the number of such codewords is often limited, which restricts the amount of reliability information available to the decoder. To address this issue, we extend the component counting mechanism from minimum-weight dual codewords to dual codewords whose Hamming weights lie within a prescribed upper bound. We further introduce a normalized component counting rule to suppress the overcounting bias caused by the much larger populations of higher-weight classes. Based on this framework, we develop a hard-decision iterative error reduction decoder and a soft information-aided iterative error reduction decoder. Numerical results show that the proposed method consistently outperforms existing minimum-weight shift-sum baselines especially for short codes where additional dual codeword information is particularly valuable.

Abstract: We study universal unequal error protection of a single special message over discrete memoryless channels. We construct a sequence of codes with a decoding rule that does not depend on the channel transition law. We derive an achievable error exponent for the special message while ensuring reliable decoding of the ordinary messages. The resulting exponent strictly improves on the natural equal-error-protection random-coding baseline and remains positive as the rate approaches capacity. For symmetric channels, it converges to the optimal red-alert exponent at capacity.

Abstract: We study the undetected-error exponent of discrete memoryless channels with erasure decoding. Assuming that the undetected-error and erasure probabilities vanish, we derive an upper bound on the undetected-error exponent at positive rates. The bound uses Somekh-Baruch's genie-aided argument where an auxiliary output provides the decoder with a list of codewords. It recovers Telatar's tight rate-zero converse and is at least as tight as the straight-line bound. A numerical example shows that the improvement over the straight-line bound can be strict.

Abstract: Symbol-pair distances of simple-root cyclic codes are important for evaluating error correction performance under overlapping pair read channels. In this paper, we construct three families of simple-root cyclic codes over finite fields of odd characteristic via generator polynomials. For length \(2q+2\), the minimum symbol-pair distance is \(10\) for every odd prime power \(q\ge 7\), established by excluding invalid support configurations. For length \(4q-4\), when \(q\equiv 3\pmod 4\) and finite field constraints hold, it is \(9\) or \(10\). For length \(2(q-1)\), reciprocal-polynomial analysis shows it is \(11\) or \(12\). All obtained codes are MDS or AMDS symbol-pair codes.

Abstract: Transmitters in adaptive communication systems may switch their modulation scheme at an unknown time, and a receiver must detect this switch quickly without raising frequent false alarms. Existing modulation classification methods label a finite block of received data and do not provide stopping rules with controlled false-alarm and delay performance. We study the sequential detection of a switch from one phase shift keying (PSK) scheme to another over an additive white Gaussian noise channel. We build cumulative-sum (CUSUM) and moving-sum (MOSUM) detectors from the log-likelihood ratio and its low-SNR approximation. For a switch from quadrature PSK to binary PSK, we derive the moment-generating function and probability density function of the approximate statistic. We use them to relate the CUSUM threshold to the average run length to false alarm and the worst-case detection delay. For a switch between any two PSK orders, we approximate the vector of overlapping MOSUM statistics as multivariate Gaussian and obtain bounds on both measures. Monte Carlo simulations show that our derived expressions for ARL2FA and WADD closely match the empirical results across thresholds, validating the accuracy of the proposed analytical approximations and theoretical bounds.

Abstract: We study the relationship between strong data-processing inequalities for Rényi divergences and hypercontractivity. For Rényi divergences of order greater than one, we characterize the contraction of Rényi divergences through a property of the dual Markov operator that we name power contraction. Using the contraction coefficient, we derive an upper bound on rectangle probabilities in terms of their marginals that is at least as strong as the corresponding bound obtained via hypercontractivity. Combining this result with inversion of the binary Rényi divergence, we derive communication lower bounds for one-way communication-assisted agreement distillation. Examples involving binary symmetric and rare-event Z-channels illustrate the improvements over standard hypercontractive converses.

Abstract: Omura's lower bound on the probability of correct decoding at rates above capacity has an exponent of sphere packing form. Dueck and Körner refined Omura's argument through a codebook extension step and obtained the exact strong converse exponent. It has been claimed in the literature that the two exponents coincide at all rates, but the argument contains an error. To settle this question, we first show that above a threshold rate, the strong converse exponent follows a straight line of unit slope determined by the Rényi capacity of order infinity. This threshold rate plays the role of a critical rate above capacity, while the order-infinity capacity plays that of a cutoff rate. We then show that the two exponents coincide up to a second threshold rate, which lies above the first, and differ strictly beyond it. Examples include modulo-additive channels, where there is no gap; and the binary erasure channel, where the thresholds and the gap are computed in closed form.

Abstract: Quasi-cyclic low-density parity-check (QC-LDPC) codes are widely used in modern communication standards due to their simple structure and efficient encoder and decoder implementations. However, the commutativity of the circulant permutation matrices used in their construction significantly limits their girth and minimum distance, affecting iterative decoding performance. In this paper, we investigate protograph-based codes constructed using permutation polynomials of higher degree, focusing on a closed family of non-commuting binomial permutation polynomials that enables these constructions to exceed both of these girth and minimum distance upper bounds while maintaining a simple representation. Simulation results demonstrate improved decoding performance compared to constructions based on affine permutation polynomials, which include QC-LDPC codes.

Abstract: HARS (Hybrid Adaptive Reliability-Aware and Syndrome-Aware) is a bit-flipping decoder that combines local channel reliability with parameter selection from the initial syndrome. A bounded check reliability modifies the parity-check weight, while the initial syndrome weight selects the base weight, reliability coefficient, threshold decay factor, and perturbation amplitude for each frame. Computing these quantities at initialization supports synchronous multi-bit updates with a simple iterative datapath. For rate-one-half PEGReg and WiMAX codes over the binary-input additive white Gaussian noise channel, HARS reduces both bit-error rate (BER) and mean iteration count relative to a calibrated SM-NGDBF baseline. The PEGReg BER at 3.25 dB is approximately one-sixth of the comparison value; the WiMAX BER at 2.75 dB is reduced by 59%. A seven-fractional-bit FPGA implementation shares 132 variable-node lanes among four frame contexts. Precomputed check weights, encoded threshold states, and parallel perturbation generation support one group per clock during sustained decoding. On an XC7A200T at 60 MHz, for preloaded 128-frame batches with a ready receiver, measured coded throughputs are 49.7 Mb/s at 2.50 dB and 361.9 Mb/s at 4.00 dB, including initialization and output transfers.

Abstract: The design of optimal causal linear feedback schemes for additive white Gaussian noise (AWGN) channels with noisy output feedback has remained an open problem for over 60 years. Prior work has focused on restricted policy classes, especially passive (uncoded) noisy output feedback, where only the transmitter performs feedback coding. However, passive noisy output feedback fundamentally lacks the degrees of freedom required to attain the information-theoretic performance limit in general. In this paper, we consider the active (coded) noisy output feedback setting, where both the transmitter and the receiver perform feedback coding. We then develop a constructive KKT-optimal active linear feedback design that asymptotically attains the Elias-Butman SNR converse bound, thereby establishing MSE/SNR optimality over the entire class of causal linear schemes. Furthermore, we prove that the optimal passive feedback solution is recovered as a special case of the active design. This passive solution admits a Geometric Toeplitz (GT) structure with a Chance-Love (CL)-style one-shot polynomial characterization, and can be computed with O(log T) complexity. Thus, our results provide an affirmative answer to the long-standing optimality question for noisy output feedback under causal linear feedback coding, and our numerical results support the theoretical findings.

Abstract: We determine the exact second-order asymptotics of covert communication over binary-input discrete memoryless channels when covertness is measured by variational distance. Previous work by Tahmasbi and Bloch [IEEE Trans. Inf. Theory, Apr. 2019] characterized the first-order asymptotics and derived achievability and converse bounds on the second-order term, but these bounds do not match. The gap arises from an additional penalty of order \(n^{1/4}\) in the achievability bound. We show that this penalty can be removed through a sharper analysis of the distribution of the warden's output induced by pulse-position modulation. Specifically, we express the variational distance through the Bhattacharyya coefficient of two distributions and the expectation of a continuous function of the log-likelihood ratio. Because the resulting expectation involves a continuous function rather than the probability of a likelihood-ratio event, an analysis of the characteristic function combined with a Gaussian smoothing argument reduces the approximation error from \(O(n^{-1/4})\) (derived from the Berry--Esseen bound in prior work) to \(O(n^{-1/2})\). With this better controlled approximation error, we manage to derive a matching achievability result to the existing converse result, thus establishing the exact second-order asymptotics.

Abstract: We show that binary codes of length $n$ correcting two deletions exist with redundancy $3\log_2n+O(\log_2\log_2n)$. The previous best upper bound had leading coefficient $4$, unchanged since 1965, while the best known lower bound has coefficient $2$. More generally, codes correcting $t\ge2$ deletions exist with redundancy $(2t-1)\log_2n+O_t(\log_2\log_2n)$, improving the coefficient $2t$. We extract a code from one label class of a random linear hash of substring counts, with about $n$ times fewer labels than a direct construction. Confusable words that still share a label are separated by a two-colouring after discarding the words in components with odd cycles, and these are few because an odd cycle forces the edits along it to overlap.

Abstract: The feedback capacity of power-constrained channels with additive stationary Gaussian noise was formulated by Kim in \cite{kim2010feedback} as an infinite-dimensional optimization over the spectrum of an independent stationary Gaussian component and a strictly causal feedback filter. Kim further asserted that the independent stationary component can be omitted. However, Derpich and Østergaard \cite{derpich2022comments} identified a gap in the original proof, leaving the original derivation of the filter-only capacity formula incomplete. In this paper, we establish this reduction at the level of capacity suprema. For the noise spectrum bounded away from zero, canonical spectral factorization represents the independent component as the defect of a Schur function from innerness. Finite Blaschke products that preserve its value at the origin produce filters with exactly the same output spectrum and convergent input powers. A power-backoff argument then enforces the original power constraint. Finally, an interleaved AR(1) example relates the reduction to the capacity-achieving SK(2) construction. The general approximation theorem does not require or establish attainment by a single filter.

Abstract: We establish two degree-dependent results for spatially coupled MacKay-Neal codes on binary-input memoryless symmetric channels. The ensembles use uniform random smoothing and shorten both variable types outside the active chain. For $r=g=3$, every integer $\ell\geq4$, and each channel of capacity greater than $R=r/\ell$, there is a sequence of code realizations whose actual transmitted rate tends to $R$ and whose average sum-product bit error tends to zero. For $r=g=2$ and each $\ell\in\{3,4,5\}$, an interval of binary symmetric channels has capacity greater than $R$ but retains positive transmitted-bit error under terminated density evolution as chain length grows relative to coupling width. Both results follow from the signs of the same density-valued potential at uncoupled fixed points. The degree-three proof combines analytic bounds for $\ell\geq33$ with exact interval certificates for $4\leq\ell\leq32$, followed by threshold saturation and a projection argument for the actual rate. The degree-two proof analytically constructs a fixed point with negative potential and controls both boundary contributions. We relate the potential and the degree-two bifurcation condition to earlier statistical-mechanical predictions. The finite certificates and verification software are available in a versioned supplement.

Abstract: 6G is expected to push ultra-reliable low-latency communication (URLLC) toward stringent residual-error targets for mission-critical services, where undetected errors and erasures carry fundamentally different costs. Block error rate (BLER) conflates block confusions (undetected errors) and block erasures, which have fundamentally different impacts on system reliability. This paper extends the confusion and erasure analysis of error-bounded decoders to square quadrature amplitude modulation (QAM) constellations in the finite blocklength (FBL) regime. To handle QAM's heterogeneous symbol energies - which make the per-pair Euclidean distance a distribution rather than a single value - we derive analytical lower and upper bounds on the block confusion rate by, respectively, collapsing this distribution to its root-mean-square (RMS) distance and averaging the pairwise confusion over it. These bounds are proven to be monotonically decreasing in both the average symbol energy and the blocklength, with the decrease rate governed by the constellation order. Numerical results confirm that as the signal-to-noise ratio (SNR) or redundancy increases, the confusion rate falls many orders of magnitude below the reliability target, leaving detectable erasures as the dominant residual error.

Abstract: We study variable-length feedback (VLF) codes over a discrete memoryless channel under average decoding-time and error-probability constraints. In the non-vanishing error probability regime, Polyanskiy, Poor, and Verdú (2011) derive achievability and converse bounds on the logarithm of the maximum achievable codebook size. These bounds establish the $ε$-capacity but leave an order-$\log N$ gap in the second-order expansion, where $N$ is the average decoding time. Yavas and Tan (2025) improve the coefficient of $\log N$ in the achievability bound from $-1$ to $-\frac{C}{C_1}$, where $C$ is the channel capacity and $C_1$ is the largest Kullback--Leibler divergence between two conditional output distributions. We derive a converse with the same coefficient, establishing the second-order fundamental limit for every positive-capacity discrete memoryless channel with finite $C_1$. The result also covers the moderate-deviations and error-exponent regimes, including polynomially decaying error probabilities. The converse uses Rényi entropy and the extrinsic Jensen--Shannon divergence. We also derive necessary properties of asymptotically optimal VLF codes. First-order-optimal codes must have an early-stopping branch, and second-order-optimal codes must additionally exhibit communication and confirmation behavior. Finally, for the binary erasure channel, we determine the exact minimum expected decoding time for every message-set size and admissible error probability.

Abstract: This paper presents an algebraic method to construct convolutional codes with guaranteed \emph{free and local minimum distances} without limit based on cyclic codes of odd lengths. The constructions are simple but effective, and no computer search is needed. For any two positive integers $r$ and $t$ with $1 \leq r < t$, a rate-$r/t$ convolutional code $\mathcal{C}_{\text{convol}}$ can be constructed by using a chain of $r$ cyclic codes $\mathcal{C}_0, \mathcal{C}_1, \ldots, \mathcal{C}_{r-1}$ of the same length $n$ which satisfy the inclusion condition, $\mathcal{C}_0 \supset \mathcal{C}_1 \supset \ldots \supset \mathcal{C}_{r-1}$. Such a convolutional code $\mathcal{C}_{\text{convol}}$ is composed of a \emph{semi-infinite chain of identical local codes} confined in a diagonal band of width $n$. Each local code $\mathcal{C}_{\text{local}}$ of $\mathcal{C}_{\text{convol}}$ is formed from the $r$ cyclic codes in the code chain and is a specially localized subcode of the \emph{mother code} $\mathcal{C}_0$ in the code chain. The minimum distance $d_{\text{local}}$ of each local code of $\mathcal{C}_{\text{convol}}$ is lower bounded by the minimum distance $d_0$ of the mother code $\mathcal{C}_0$ in the code chain. The local structure of $\mathcal{C}_{\text{convol}}$ allows it to be decoded based on a designed parity-check matrix of the mother code $\mathcal{C}_0$ using a sliding window decoding scheme.

Abstract: We characterize the degree of freedom (DoF) of a point-to-point blockwise memoryless channel without prior channel state information (CSI), with a fixed number $K$ of propagation paths, where the transmitter (Tx) and the receiver (Rx) are equipped with nonuniform linear arrays (NULAs) of $N_t$ and $N_r$ antennas, respectively. The positions of array elements are fixed, known, pairwise distinct, and need not be equally spaced. The uniform linear array (ULA) is a special case. In each block of length $T$, the continuous angles of arrival (AoAs), angles of departure (AoDs), and independent complex Gaussian path gains are redrawn. Both Tx and Rx know the state distributions but are not given the current realizations before transmission. The receiver may estimate the channel from reference signals or decode without explicit channel estimation, with reference symbols counted in $T$ and their energy counted against the power constraint. Under the aforementioned model, we show that the DoF is $1-\frac{1}{T}$ for $K=1$, and $K(1-\frac{3}{2T})$ for $K \geq 2$, when $N_r\ge K+1$, $N_t\ge\max\{K,2\}$, and $T\ge K$. The analytical results are further demonstrated by their applications to the DoF tradeoff analysis in integrated sensing and communication (ISAC). For more general array structures, an achievability result is established, while the converse remains open in general.

Abstract: Recurrent detectors such as bidirectional long short-term memory (Bi-LSTM) networks are low-complexity alternatives to the optimal Bahl-Cocke-Jelinek-Raviv (BCJR) detector for faster-than-Nyquist (FTN) signaling. Motivated by convolutional detectors that build the intersymbol interference (ISI) structure into their architecture, we ask whether processing nested ISI windows in separate recurrent branches improves the bit error rate (BER) of a Bi-LSTM. Across roughly 260 controlled trainings it does not: at a matched parameter budget and a matched readout, the multi-window architecture never significantly beats a plain Bi-LSTM. Nested windowing is an invertible rearrangement that adds no information, extra branches only add bottlenecks, and a distillation diagnostic shows the network is already near optimal for its window. The limitation is therefore the observation model, not the architecture. Keeping the architecture fixed, we pre-whiten the input, restoring the conditional independence that colored matched-filter noise violates, and distill the BCJR soft posterior into the network. With 3.4% more parameters this reaches 1.05 times the BCJR BER at a compression factor of 0.8 and 1.89 times at 0.7, improving to 1.47 times when the whitened window is widened. The 23.7% BER reduction at 0.8 requires an ill-conditioned ISI matrix but is not monotone in the conditioning, and it holds across five independent noise realizations and a symbol-level McNemar test.