arXiv subjects

Guiying Yan

Publications and source records attributed to Guiying Yan.

At least 55 records · Page 3Linked to original sources

arXiv2024

Partial Orders in Rate-Matched Polar Codes

In this paper, we establish the partial order (POs) for both the binary erasure channel (BEC) and the binary memoryless symmetric channel (BMSC) under any block rate-matched polar codes. Firstly, we define the POs in the sense of rate-matched polar codes as a sequential block version. Furthermore, we demonstrate the persistence of POs after block rate matching in the BEC. Finally, leveraging the existing POs in the BEC, we obtain more POs in the BMSC under block rate matching. Simulations show that the PW sequence constructed from β-expansion can be improved by the tool of POs. Actually, any fixed reliable sequence in the mother polar codes can be improved by POs for rate matching.

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arXiv2024

A Method to Reduce the Complexity of Computing the Complete Weight Distribution of Polar Codes

The code spectrum of polar codes is crucial to the performance of polar codes. Based on the lower-triangular affine group (LTA) of decreasing monomial codes and the one-variable descendance (ovd) relation, we define a new subgroup of LTA which can find more cosets with the same weight distribution. Using this algebraic structure, we further reduce the complexity by proofing the group action on a coset set is transitive. Our method is an enhanced version of previous research, and the complexity of most cases can be reduced exceeding several times.

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arXiv2024

Achievability Bounds on Unequal Error Protection Codes

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.

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arXiv2024

An Optimization Model for Offline Scheduling Policy of Low-density Parity-check Codes

In this study, an optimization model for offline scheduling policy of low-density parity-check (LDPC) codes is proposed to improve the decoding efficiency of the belief propagation (BP). The optimization model uses the number of messages passed (NMP) as a metric to evaluate complexity, and two metrics, average entropy (AE), and gap to maximum a posteriori (GAP), to evaluate BP decoding performance. Based on this model, an algorithm is proposed to optimize the scheduling sequence for reduced decoding complexity and superior performance compared to layered BP. We validated the proposed algorithm on LDPC codes constructed following 5G New Radio, which resulted in a reduction of decoding complexity of more than 20$\%$ compared to LBP.

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arXiv2024

Second-Order Identification Capacity of AWGN Channels

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 blocklength.

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arXiv2024

On automorphism groups of polar codes

Over the past years, Polar codes have arisen as a highly effective class of linear codes, equipped with a decoding algorithm of low computational complexity. This family of codes share a common algebraic formalism with the well-known Reed-Muller codes, which involves monomial evaluations. As useful algebraic codes, more specifically known as decreasing monomial codes, a lot of decoding work has been done on Reed-Muller codes based on their rich code automorphisms. In 2021, a new permutation group decoder, referred to as the automorphism ensemble (AE) decoder, was introduced. This decoder can be applied to Polar codes and has been shown to produce similar decoding effects. However, identifying the right set of code automorphisms that enhance decoding performance for Polar codes remains a challenging task. This paper aims to characterize the full automorphism group of Polar codes. We will prove a reduction theorem that effectively reduces the problem of determining the full automorphism group of arbitrary random Polar codes to that of a specified class of Polar codes. Besides, we give exact classification of the full automorphism groups of families of Polar codes that are constructed using the Reed-Muller codes.

math.CO↗

arXiv2024

Planar Turán number for balanced double stars

Planar Turán number, denoted by $ex_{\mathcal{P}}(n,H)$, is the maximum number of edges in an $n$-vertex planar graph which does not contain $H$ as a subgraph. Ghosh, Győri, Paulos and Xiao initiated the topic of the planar Turán number for double stars. For balanced double star, $S_{3,3}$ is the only remaining graph need to be considered. In this paper, we give the exact value of $ex_{\mathcal{P}}(n,S_{3,3})$, forcing the planar Turán number for all balanced double stars completely determined.

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arXiv2024

Anti-Ramsey Numbers of Expansions of Doubly Edge-critical Graphs in Uniform Hypergraphs

For an $r$-graph $H$, the anti-Ramsey number ${\rm ar}(n,r,H)$ is the minimum number $c$ of colors such that for any edge-coloring of the complete $r$-graph on $n$ vertices with at least $c$ colors, there is a copy of $H$ whose edges have distinct colors. A 2-graph $F$ is doubly edge-$p$-critical if the chromatic number $χ(F - e)\geq p$ for every edge $e$ in $F$ and there exist two edges $e_1,e_2$ in $F$ such that $χ(F -e_1- e_2)=p-1$. The anti-Ramsey numbers of doubly edge-$p$-critical 2-graphs were determined by Jiang and Pikhurko \cite{Jiang&Pikhurko2009}, which generalized the anti-Ramsey numbers of cliques determined by Erdős, Simonovits and Sós \cite{Erdos&Simonovits&Sos1975}. In general, few exact values of anti-Ramsey numbers of $r$-graphs are known for $r\geq 3$. Given a 2-graph $F$, the expansion $F^{(r)}$ of $F$ is an $r$-graph on $|V(F)|+(r-2)|F|$ vertices obtained from $F$ by adding $r-2$ new vertices to each edge of $F$. In this paper, we determine the exact value of ${\rm ar}(n,r,F^{(r)})$ for any doubly edge-$p$-critical 2-graph $F$ with $p>r\geq 3$ and sufficiently large $n$.

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arXiv2024

Anti-Ramsey numbers of loose paths and cycles in uniform hypergraphs

For a fixed family of $r$-uniform hypergraphs $\mathcal{F}$, the anti-Ramsey number of $\mathcal{F}$, denoted by $ ar(n,r,\mathcal{F})$, is the minimum number $c$ of colors such that for any edge-coloring of the complete $r$-uniform hypergraph on $n$ vertices with at least $c$ colors, there is a rainbow copy of some hypergraph in $\mathcal{F}$. Here, a rainbow hypergraph is an edge-colored hypergraph with all edges colored differently. Let $\mathcal{P}_k$ and $\mathcal{C}_k$ be the families of loose paths and loose cycles with $k$ edges in an $r$-uniform hypergraph, respectively. In this paper, we determine the exact values of $ ar(n,r,\mathcal{P}_k)$ and $ ar(n,r,\mathcal{C}_k)$ for all $k\geq 4$ and $r\geq 3$.

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arXiv2024

On the Weight Distribution of Weights Less than $2w_{\min}$ in Polar Codes

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 $2w_{\min}$, where $w_{\min}$ represents 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 $2w_{\min}$. Moreover, the proposed enumeration algorithm runs in polynomial time with respect to the code length.

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arXiv2024

New Partial Orders of Polar Codes for BMSC

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). %Additionally, we discover more special POs in the Binary Symmetric Channel (BSC). We provide examples that demonstrate the inability of existing POs to deduce these novel POs. Furthermore, we establish upper bounds for the expansion parameter $β$ if the polar codes constructed by $β$-expansion method obey these POs.

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arXiv2023

On the Weight Spectrum Improvement of Pre-transformed Reed-Muller Codes and Polar Codes

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.

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arXiv2023

Theoretical Bounds for the Size of Elementary Trapping Sets by Graphic Methods

Elementary trapping sets (ETSs) are the main culprits for the performance of LDPC codes in the error floor region. Due to the large quantity, complex structures, and computational difficulties of ETSs, how to eliminate dominant ETSs in designing LDPC codes becomes a pivotal issue to improve the error floor behavior. In practice, researchers commonly address this problem by avoiding some special graph structures to free specific ETSs in Tanner graph. In this paper, we deduce the accurate Turán number of $θ(1,2,2)$ and prove that all $(a,b)$-ETSs in Tanner graph with variable-regular degree $d_L(v)=γ$ must satisfy the bound $b\geq aγ-\frac{1}{2}a^2$, which improves the lower bound obtained by Amirzade when the girth is 6. For the case of girth 8, by limiting the relation between any two 8-cycles in the Tanner graph, we prove a similar inequality $b\geq aγ-\frac{a(\sqrt{8a-7}-1)}{2}$. The simulation results show that the designed codes have good performance with lower error floor over additive white Gaussian noise channels.

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arXiv2023

Improving the Gilbert-Varshamov Bound by Graph Spectral Method

We improve Gilbert-Varshamov bound by graph spectral method. Gilbert graph $G_{q,n,d}$ is a graph with all vectors in $\mathbb{F}_q^n$ as vertices where two vertices are adjacent if their Hamming distance is less than $d$. In this paper, we calculate the eigenvalues and eigenvectors of $G_{q,n,d}$ using the properties of Cayley graph. The improved bound is associated with the minimum eigenvalue of the graph. Finally we give an algorithm to calculate the bound and linear codes which satisfy the bound.

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arXiv2023

On the Message Passing Efficiency of Polar and Low-Density Parity-Check Decoders

This study focuses on the efficiency of message-passing-based decoding algorithms for polar and low-density parity-check (LDPC) codes. Both successive cancellation (SC) and belief propagation (BP) decoding algorithms are studied {in} the message-passing framework. Counter-intuitively, SC decoding demonstrates the highest decoding efficiency, although it was considered a weak decoder {in terms of} error-correction performance. We analyze the complexity-performance tradeoff to dynamically track the decoding efficiency, where the complexity is measured by the number of messages passed (NMP), and the performance is measured by the statistical distance to the maximum a posteriori (MAP) estimate. This study offers a new insight into the contribution of each message passed in decoding, and compares various decoding algorithms on a message-by-message level. The analysis corroborates recent results on terabits-per-second polar SC decoders, and might shed light on better scheduling strategies.

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arXiv2023

Rainbow Hamilton cycle in hypergraph system

In this paper, we develop a new rainbow Hamilton framework, which is of independent interest, settling the problem proposed by Gupta, Hamann, Müyesser, Parczyk, and Sgueglia when $k=3$, and draw the general conclusion for any $k\geq3$ as follows. A $k$-graph system $\textbf{H}=\{H_i\}_{i\in[n]}$ is a family of not necessarily distinct $k$-graphs on the same $n$-vertex set $V$, moreover, a $k$-graph $H$ on $V$ is rainbow if $E(H)\subseteq \bigcup_{i\in[n]}E(H_i)$ and $|E(H)\cap E(H_i)|\leq1$ for $i\in[n]$. We show that given $γ> 0$, sufficiently large $n$ and an $n$-vertex $k$-graph system $\textbf{H}=\{H_i\}_{i\in[n]}$ , if $δ_{k-2}(H_i)\geq(5/9+γ)\binom{n}{2}$ for $i\in[n]$ where $k\geq3$, then there exists a rainbow tight Hamilton cycle. This result implies the conclusion in a single graph, which was proved by Lang and Sanhueza-Matamala [$J. Lond. Math. Soc., 2022$], Polcyn, Reiher, Rödl and Schülke [$J. Combin. Theory \ Ser. B, 2021$] independently.

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arXiv2022

Deterministic Identification over Channels without CSI

Identification capacities of randomized and deterministic identification were proved to exceed channel capacity for Gaussian channels \emph{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.

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arXiv2022

The Complete SC-invariant Affine Automorphisms of Polar Codes

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.

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