The weight spectrum plays a crucial role in the performance of error-correcting codes. Pre-transformation with an upper-triangular matrix improves the weight spectrum of polar codes while retaining polarization. However, a theoretical analysis to quantify the improvement for pre-transformed rate-compatible polar codes is missing. In this paper, we calculate the average spectrum of random upper-triangular pre-transformed shortened and punctured polar codes. Our approach tran-scends the limitations imposed by partial ordering and specific rate matching patterns. A key feature of our approach is its polynomial complexity in relation to the code length, making it computationally feasible. Simulation results affirm that our findings provide an accurate approximation on the performance of pre-transformed rate-compatible polar codes.

In this paper, we establish the partial orders (POs) of rate-compatible polar codes under both the binary erasure channel (BEC) and the binary memoryless symmetric channel (BMSC). Firstly, we define the POs for rate-compatible polar codes under block rate matching. Additionally, we demonstrate that certain POs for mother code lengths remain valid under rate matching in the BEC. Finally, leveraging the existing POs in the BEC, we derive POs in the BMSC under block rate matching.

The weight spectrum plays a crucial role in the performance of error-correcting codes. Despite substantial theoretical exploration into polar codes with mother code length, a framework for the weight spectrum of rate-compatible polar codes remains elusive. In this paper, we address this gap by enumerating the number of minimum-weight codewords for quasi-uniform punctured, Wang-Liu shortened, and bit-reversal shortened decreasing polar codes. Notably, our algorithms operate with polynomial complexity relative to the code length. Simulation results affirm that our discoveries provide an accurate approximation on the performance of rate-compatible polar codes.

In this paper, we study the lossless analog compression fori.i.d.discrete-continuous mixed signals via the polarization-based framework. We prove that for discrete-continuous mixed source, the error probability of maximum a posteriori (MAP) estimation polarizes under the Hadamard transform, which extends the polarization phenomenon to analog domain. Building on this insight, we propose the partial Hadamard compression and develop the corresponding analog successive cancellation (SC) decoder. The proposed scheme consists of deterministic measurement matrices and non-iterative reconstruction algorithm, providing benefits in both space and computational complexity. Using the polarization of error probability, we prove that our approach achieves the information-theoretical limit for lossless analog compression developed by Wu and Verdú.

In this paper, we establish the second-order randomized identification capacity (RID capacity) of the Additive White Gaussian Noise Channel (AWGNC). On the one hand, we obtain a refined version of Hayashi's theorem to prove the achievability part. On the other, we investigate the relationship between identification and channel resolvability, then we propose a finer quantization method to prove the converse part. Consequently, the second-order RID capacity of the AWGNC has the same form as the second-order transmission capacity. The only difference is that the maximum number of messages in RID scales double exponentially in the block length.

In this paper, we define partial orders (POs) of polar codes based on the Bhattacharyya parameter and the bit-error probability, respectively. These POs are applicable to arbitrary binary memoryless symmetric channel (BMSC). Leveraging the extremal inequalities of polarization transformation, we derive new POs for BMSC based on the corresponding POs observed in the Binary Erasure Channel (BEC). We provide examples that demonstrate the inability of existing POs to deduce these novel POs. Furthermore, we establish upper bounds for the expansion parameter$\beta$if the polar codes constructed by$\beta- \mathbf{expansion}$method obey these POs.

This paper proposes a reduced complexity ordered statistics decoding (OSD) algorithm for linear block codes, the namely order skipping (OS)-OSD algorithm. An approximated correlation distance lower bound (CDLB) is derived by utilizing likelihood of the received symbols over the least reliable positions (LRPs). It enables the assessment of whether the higher-order decoding can yield a more likely codeword estimation. If not, they can be skipped. Error-correction performance of the OSOSD is analyzed. In particular, the decoding error probability of OS-OSD with order one is theoretically characterized. Our simulation results verify that the OS-OSD can achieve a significant complexity reduction over the state-of-the-art OSD without compromising the decoding performance.

Unequal error protection (UEP) codes can facilitate the transmission of messages with different protection levels. In this paper, we study the achievability bounds on UEP by the generalization of Gilbert-Varshamov (GV) bound. For the first time, we show that under certain conditions, UEP enhances the code rate comparing with time-sharing (TS) strategies asymptotically.

The number of low-weight codewords is critical to the performance of error-correcting codes. In 1970, Kasami and Tokura characterized the codewords of Reed-Muller (RM) codes whose weights are less than 2wmin, wherewminrepresents the minimum weight. In this paper, we extend their results to decreasing polar codes. We present the closed-form expressions for the number of codewords in decreasing polar codes with weights less than 2wmin. Moreover, the proposed enumeration algorithm runs in polynomial time with respect to the code length.

The automorphism ensemble (AE) decoding framework for polar codes attracts much attention recently. It decodes multiple permuted codewords with successive cancellation (SC) decoders in parallel and hence has lower latency compared to successive cancellation list (SCL) decoding. However, the AE decoding framework is ineffective for permutations falling into the lower-triangular affine (LTA) automorphism group, as they are invariant under SC decoding. Therefore, the block lower-triangular affine (BLTA) group was discovered to achieve better AE decoding performance. However, the equivalence of the BLTA group and the complete affine automorphism group was unresolved. Additionally, some automorphisms in BLTA group are also SC-invariant, thus are redundant in AE decoding. In this paper, we prove that BLTA group coincides with the complete automorphisms of decreasing polar codes that can be formulated as affine transformations. Also, we find a necessary and sufficient condition related to the block lower-triangular structure of transformation matrices to identify SC-invariant automorphisms. Furthermore, We present an algorithm that efficiently identifies all SC-invariant affine automorphisms under specific constructions.

In this paper, we study the lossless analog compression for i.i.d. nonsingular signals. Through analyzing analog polarization under Hadamard transform, we propose efficient successive cancellation (SC) decoding algorithm over analog domain. Thanks to the polarization of Rényi information dimension (RID) and the absorption of discrete entropy, we prove that the proposed scheme achieves the information-theoretical limit for lossless analog compression developed by Wu and Verdú.

Pre-transformation with an upper-triangular matrix (including cyclic redundancy check (CRC), parity-check (PC) and polarization-adjusted convolutional (PAC) codes) improves the weight spectrum of Reed-Muller (RM) codes and polar codes significantly. However, a theoretical analysis to quantify the improvement is missing. In this paper, we provide asymptotic analysis on the number of low-weight codewords of the original and pre-transformed RM codes respectively, and prove that pre-transformation significantly reduces low-weight codewords, even in the order sense. For polar codes, we prove that the average number of minimum-weight codewords does not increase after pre-transformation. Both results confirm the advantages of pre-transformation.

The rapid development of Internet of Things (IoT) requires new massive random access protocols to support the massive devices. Recently, Polyanskiy [1] established an information-theoretic formulation of unsourced multiple access (uMAC) problem and proposed theoretical finite-length achievability and converse bounds. In this paper, we further study the tradeoff between the number of active users and the energy-per-bit of random Gaussian codebook under maximum likelihood decoding and use two methods to improve the finite-length achievability bounds when the number of users is large and small, respectively. Our new results improve the finite-length achievability bound by more than 0.15 dB when per-user probability of error (PUPE) is 10−1, and more than 0.25 dB when PUPE is 10−3.

Automorphism ensemble (AE) decoding for polar codes was proposed by decoding permuted codewords with successive cancellation (SC) decoders in parallel and hence has lower latency compared to that of successive cancellation list (SCL) decoding. However, some automorphisms are SC-invariant, thus are redundant in AE decoding. In this paper, we find a necessary and sufficient condition related to the block lower-triangular structure of transformation matrices to identify SC-invariant automorphisms. Furthermore, we provide an algorithm to determine the complete SC-invariant affine automorphisms under a specific polar code construction.

Identification capacities of randomized and deterministic identification were proved to exceed channel capacity for Gaussian channels with channel side information (CSI). In this work, we extend deterministic identification to the block fading channels without CSI by applying identification codes for both channel estimation and user identification. We prove that identification capacity is asymptotically higher than transmission capacity even in the absence of CSI. And we also analyze the finite-length performance theoretically and numerically. The simulation results verify the feasibility of the proposed blind deterministic identification in finite blocklength regime.

Recently, a permutation-based successive cancellation (PSC) decoding framework for polar codes attracts much attention. It decodes several permuted codewords with indepen-dent successive cancellation (SC) decoders. Its latency thus can be reduced to that of SC decoding. However, the PSC framework is ineffective for permutations falling into the lower-triangular affine (LTA) automorphism group, as they are invariant under SC decoding. As such, a larger block lower-triangular affine (BLTA) group that contains SC-variant permutations was discovered for decreasing polar codes. But it was unknown whether BLTA equals the complete automorphism group. In this paper, we prove that BLTA equals the complete automorphisms of decreasing polar codes that can be formulated as affine transformations.

Polar codes are the first class of channel codes achieving the symmetric capacity of the binary-input discrete memoryless channels (B-DMC) with efficient encoding and decoding algorithms. But the weight spectrum of polar codes is relatively poor compared to Reed-Muller (RM) codes, which degrades their maximum-likehood (ML) performance. Pre-transformation with an upper-triangular matrix (including cyclic redundancy check (CRC), parity-check (PC) and polarization-adjusted convolutional (PAC) codes), improves weight spectrum while retaining polarization. In this paper, the weight spectrum of upper-triangular pre-transformed polar codes is mathematically analyzed. In particular, we focus on calculating the number of low-weight codewords due to their impact on error-correction performance. Simulation results verify the accuracy of the analysis.