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To address the challenge of constructing short girth-8 quasi-cyclic (QC) low-density parity-check (LDPC) codes, a novel construction framework based on vertical symmetry (VS) is proposed. Basic properties of the VS structure are presented. With the aid of these properties, existing explicit constructions for column weights from three to five which can be transformed into the VS structure are sorted out. Then two novel explicit constructions with the VS structure which guarantee short codes are presented for column weights of three and six. Moreover, an efficient search-based method is also proposed to find short codes with the VS structure. Compared with the state-of-the-art benchmarks, both the explicit constructions and the search-based method presented in this paper can provide shorter codes for most cases. Simulation results show that the new shorter codes can perform almost the same as or better than the longer existing counterparts. Thus, the new shorter codes can fit better with the low-latency requirement for modern communication systems., Comment: 17 pages, 5 figures; This paper has been accepted by IEEE ISIT2024

In this paper we generalize [BS22, Main Theorem] from actions of a single transformation to amenable group actions, which answers affirmatively the question raised in [BS22] by Burguet and the first-named author of the paper., Comment: 16 pages

In this paper we prove the tail variational principle for actions of countable amenable groups. This allows us to extend some characterizations of asymptotic $h$-expansiveness from $\mathbb{Z}$-actions to actions of countable amenable groups., Comment: to appear in "Israel Journal of Mathematics" (a special volume in honor of Professor Benjamin Weiss)

The paper offers a thorough study of multiorders and their applications to measure-preserving actions of countable amenable groups. By a~{\em multiorder} on a~countable group we mean any probability measure $\nu$ on the collection $\tilde{\mathcal{O}}$ of linear orders of type $\mathbb Z$ on $G$, invariant under the natural action of $G$ on such orders. Every free measure-preserving $G$-action $(X,\mu,G)$ has a~multiorder $(\tilde{\mathcal{O}},\nu,G)$ as a factor and has the same orbits as the $\mathbb Z$-action $(X,\mu,S)$, where $S$ is the \emph{successor map} determined by the multiorder factor. Moreover, the sub-sigma-algebra $\Sigma_{\tilde{\mathcal{O}}}$ associated with the multiorder factor is invariant under $S$, which makes the corresponding $\mathbb Z$-action $(\tilde{\mathcal{O}},\nu,\tilde S)$ a factor of $(X,\mu,S)$. We prove that the entropy of any $G$-process generated by a finite partition of $X$, conditional with respect to $\Sigma_{\tilde{\mathcal{O}}}$, is preserved by the orbit equivalence with $(X,\mu,S)$. Furthermore, this entropy can be computed in terms of the so-called random past, by a formula analogous to $ h(\mu,T,\mathcal P)=H(\mu,\mathcal P|\mathcal{P}^-)$ known for $\mathbb Z$-actions. The above fact is then applied to prove a variant of a result by Rudolph and Weiss. The original theorem states that orbit equivalence between free actions of countable amenable groups preserves conditional entropy with respect to a~sub-sigma-algebra $\Sigma$, as soon as the ``orbit change'' is measurable with respect to $\Sigma$. In our variant, we replace the measurability assumption by a~simpler one: $\Sigma$ should be invariant under both actions and the actions on the resulting factor should be free. In conclusion we provide a characterization of the Pinsker sigma-algebra of any $G$-process in terms of an appropriately defined remote past arising from a multiorder., Comment: 36 pages, 2 figures, Changes: slightly changed formulation and proof of Theorem 7.4, some remarks added

In this survey we will present the symbolic extension theory in topological dynamics, which was built over the past twenty years., Comment: to appear in "Acta Mathematica Sinica, English Series". arXiv admin note: text overlap with arXiv:1901.01457

We consider a fully-loaded ground wireless network supporting unmanned aerial vehicle (UAV) transmission services. To enable the overload transmissions to a ground user (GU) and a UAV, two transmission schemes are employed, namely non-orthogonal multiple access (NOMA) and relaying, depending on whether or not the GU and UAV are served simultaneously. Under the assumption of the system operating with infinite blocklength (IBL) codes, the IBL throughputs of both the GU and the UAV are derived under the two schemes. More importantly, we also consider the scenario in which data packets are transmitted via finite blocklength (FBL) codes, i.e., data transmission to both the UAV and the GU is performed under low-latency and high reliability constraints. In this setting, the FBL throughputs are characterized again considering the two schemes of NOMA and relaying. Following the IBL and FBL throughput characterizations, optimal resource allocation designs are subsequently proposed to maximize the UAV throughput while guaranteeing the throughput of the cellular user.Moreover, we prove that the relaying scheme is able to provide transmission service to the UAV while improving the GU's performance, and that the relaying scheme potentially offers a higher throughput to the UAV in the FBL regime than in the IBL regime. On the other hand, the NOMA scheme provides a higher UAV throughput (than relaying) by slightly sacrificing the GU's performance., Comment: 13 pages, 13 figures

As a typical example of bandwidth-efficient techniques, bit-interleaved coded modulation with iterative decoding (BICM-ID) provides desirable spectral efficiencies in various wireless communication scenarios. In this paper, we carry out a comprehensive investigation on tail-biting (TB) spatially coupled protograph (SCP) low-density parity-check (LDPC) codes in BICM-ID systems. Specifically, we first develop a two-step design method to formulate a novel type of constellation mappers, referred to as labeling-bit-partial-match (LBPM) constellation mappers, for SC-P-based BICM-ID systems. The LBPM constellation mappers can be seamlessly combined with high-order modulations, such as M-ary phase-shift keying (PSK) and M-ary quadrature amplitude modulation (QAM). Furthermore, we conceive a new bit-level interleaving scheme, referred to as variable node matched mapping (VNMM) scheme, which can substantially exploit the structure feature of SC-P codes and the unequal protection-degree property of labeling bits to trigger the wave-like convergence for TB-SC-P codes. In addition, we propose a hierarchical extrinsic information transfer (EXIT) algorithm to predict the convergence performance (i.e., decoding thresholds) of the proposed SC-P-based BICM-ID systems. Theoretical analyses and simulation results illustrate that the LBPM-mapped SC-P-based BICM-ID systems are remarkably superior to the state-of-the-art mapped counterparts. Moreover, the proposed SC-P-based BICM-ID systems can achieve even better error performance with the aid of the VNMM scheme. As a consequence, the proposed LBPM constellation mappers and VNMM scheme make the SC-P-based BICM-ID systems a favorable choice for the future-generation wireless communication systems., Comment: Accepted by IEEE Transactions on Vehicular Technology, Oct 2019

In this paper, we study discrete spectrum of invariant measures for countable discrete amenable group actions. We show that an invariant measure has discrete spectrum if and only if it has bounded measure complexity. We also prove that, discrete spectrum can be characterized via measure-theoretic complexity using names of a partition and the Hamming distance, and it turns out to be equivalent to both mean equicontinuity and equicontinuity in the mean.

Symbolic Extension Entropy Theorem (SEET) describes the possibility of a lossless digitalization of a dynamical system by extending it to a subshift. It gives an estimate on the entropy of symbolic extensions (and the necessary number of symbols). Unlike in the measure-theoretic case, where Kolmogorov--Sinai entropy is the estimate, in the topological setup the task reaches beyond the classical theory of entropy. Tools from an extended theory of entropy structures are needed. The main goal of this paper is to prove the SEET for actions of countable amenable groups: Let a countable amenable group $G$ act by homeomorphisms on a compact metric space $X$ and let $\mathcal M_G(X)$ denote the simplex of $G$-invariant probability measures on $X$. A function $E $ on $\mathcal M_G(X)$ equals the extension entropy function $h^\pi$ of a symbolic extension $\pi:(Y,G)\to (X,G)$, where $h^\pi(\mu)=\sup\{h_\nu(Y,G): \nu\in\pi^{-1}(\mu)\}$ ($\mu\in\mathcal M_G(X)$), if and only if $E $ is an affine superenvelope of the entropy structure of $(X,G)$. The statement is preceded by presentation of the concepts of an entropy structure and superenvelopes, adapted from $\mathbb Z$-actions. In full generality we prove a slightly weaker version of SEET, in which symbolic extensions are replaced by quasi-symbolic extensions, i.e., extensions in form of a joining of a subshift with a zero-entropy tiling system. The notion of a tiling system is a subject of earlier works and in this paper we review and complement the theory developed there. The full version of the SEET is proved for groups which are either residually finite or enjoy the comparison property. In order to describe the range of our theorem, we devote a large portion of the paper to the comparison property. Our main result in this aspect shows that all subexponential groups have the comparison property (and thus satisfy the SEET)., Comment: 89 pages. arXiv admin note: text overlap with arXiv:1712.05129

In this paper, we study dynamics of maps on quasi-graphs characterizing their invariant measures. In particular, we prove that every invariant measure of quasi-graph map with zero topological entropy has discrete spectrum. Additionally, we obtain an analog of Llibre-Misiurewicz's result relating positive topological entropy with existence of topological horseshoes. We also study dynamics on dendrites and show that if a continuous map on a dendrite, whose set of all endpoints is closed and has only finitely many accumulation points, has zero topological entropy, then every invariant measure supported on an orbit closure has discrete spectrum., Comment: 20 pages, 2 figures; substantial revision; previous title "On dynamics of quasi-graph maps"

A novel method guaranteeing nondecreasing girth is presented for constructing longer low-density parity-check (LDPC) codes from shorter ones. The parity-check matrix of a shorter base code is decomposed into N (N>=2) non-overlapping components with the same size. Then, these components are combined together to form the parity-check matrix of a longer code, according to a given N*N Latin square. To illustrate this method, longer quasi-cyclic (QC) LDPC codes are obtained with girth at least eight and satisfactory performance, via shorter QC-LDPC codes with girth eight but poor performance. The proposed method naturally includes several well-known methods as special cases, but is much more general compared with these existing approaches., Comment: 7 pages, 2 figures

Let a countable amenable group $G$ act on a \zd\ compact metric space $X$. For two clopen subsets $\mathsf A$ and $\mathsf B$ of $X$ we say that $\mathsf A$ is \emph{subequivalent} to $\mathsf B$ (we write $\mathsf A\preccurlyeq \mathsf B$), if there exists a finite partition $\mathsf A=\bigcup_{i=1}^k \mathsf A_i$ of $\mathsf A$ into clopen sets and there are elements $g_1,g_2,\dots,g_k$ in $G$ such that $g_1(\mathsf A_1), g_2(\mathsf A_2),\dots, g_k(\mathsf A_k)$ are disjoint subsets of $\mathsf B$. We say that the action \emph{admits comparison} if for any clopen sets $\mathsf A, \mathsf B$, the condition, that for every $G$-invariant probability measure $\mu$ on $X$ we have the sharp inequality $\mu(\mathsf A)<\mu(\mathsf B)$, implies $\mathsf A\preccurlyeq \mathsf B$. Comparison has many desired consequences for the action, such as the existence of tilings with arbitrarily good F{\o}lner properties, which are factors of the action. Also, the theory of symbolic extensions, known for $\mathbb z$-actions, extends to actions which admit comparison. We also study a purely group-theoretic notion of comparison: if every action of $G$ on any zero-dimensional compact metric space admits comparison then we say that $G$ has the \emph{comparison property}. Classical groups $\mathbb z$ and $\mathbb z^d$ enjoy the comparison property, but in the general case the problem remains open. In this paper we prove this property for groups whose every finitely generated subgroup has subexponential growth., Comment: overlapped by arXiv:1901.01457, and so will not be published separately (note that we have not changed the file)