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.

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.

Elementary trapping sets (ETSs) are the principal culprits for the performance of LDPC codes in the error floor region. Due to their large quantity, intricate structures, and high computational complexity, determining how to eliminate dominant ETSs in the design of LDPC codes has become a critical issue in improving error floor behavior. In this paper, we address this problem by avoiding particular theta graphs$(\theta(1,2,2)$and$\theta(2,2,2))$in the Tanner graph to eliminate specific ETSs. These can be characterized by a pivotal tool in graph theory - Turán numbers. Theoretically, we derive the exact Turán number for$\theta(1,2,2)$and demonstrate that all$(a, b)$-ETSs in a Tanner graph with variable-reaular degree$d_{L}(v)=\gamma$must satisfy the inequality$b\geq a\gamma-\frac{1}{2}a^{2}$. This result improves the lower bound previously obtained by Amirzade when the girth is 6. For girth 8, by constraining the relationship between any two 8-cycles in the Tanner graph, we establish a similar inequality$b\geq a\gamma-\frac{a(\sqrt{8a-7}-1)}{2}$. Our simulation results indicate that codes designed with these considerations exhibit improved performance and a lower error floor over additive white Gaussian noise channels.

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.

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.

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.