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In this paper we focus on modules over a finite chain ring $\mathcal{R}$ of size $q^s$. We compute the density of free modules of $\mathcal{R}^n$, where we separately treat the asymptotics in $n,q$ and $s$. In particular, we focus on two cases: one where we fix the length of the module and one where we fix the rank of the module. In both cases, the density results can be bounded by the Andrews-Gordon identities. We also study the asymptotic behaviour of modules generated by random matrices over $\mathcal{R}$. Since linear codes over $\mathcal{R}$ are submodules of $\mathcal{R}^n$ we get direct implications for coding theory. For example, we show that random codes achieve the Gilbert-Varshamov bound with high probability.

For graphs $G$ and $H$, we say that $G$ is $H$-free if it does not contain $H$ as an induced subgraph. Already in the early 1980s Alekseev observed that if $H$ is connected, then the \textsc{Max Weight Independent Set} problem (MWIS) remains \textsc{NP}-hard in $H$-free graphs, unless $H$ is a path or a subdivided claw, i.e., a graph obtained from the three-leaf star by subdividing each edge some number of times (possibly zero). Since then determining the complexity of MWIS in these remaining cases is one of the most important problems in algorithmic graph theory. A general belief is that the problem is polynomial-time solvable, which is witnessed by algorithmic results for graphs excluding some small paths or subdivided claws. A more conclusive evidence was given by the recent breakthrough result by Gartland and Lokshtanov [FOCS 2020]: They proved that MWIS can be solved in quasipolynomial time in $H$-free graphs, where $H$ is any fixed path. If $H$ is an arbitrary subdivided claw, we know much less: The problem admits a QPTAS and a subexponential-time algorithm [Chudnovsky et al., SODA 2019]. In this paper we make an important step towards solving the problem by showing that for any subdivided claw $H$, MWIS is polynomial-time solvable in $H$-free graphs of bounded degree.

Let A(n, d) denote the maximum number of codewords in a binary code of length n and minimum Hamming distance d. Deriving upper and lower bounds on A(n, d) have been a subject for extensive research in coding theory. In this paper, we examine upper and lower bounds on A(n, d) in the high-minimum distance regime, in particular, when $d = n/2 - \Theta(\sqrt{n})$. We will first provide a lower bound based on a cyclic construction for codes of length $n= 2^m -1$ and show that $A(n, d= n/2 - 2^{c-1}\sqrt{n}) \geq n^c$, where c is an integer with $1 \leq c \leq m/2-1$. With a Fourier-analytic view of Delsarte's linear program, novel upper bounds on $A(n, n/2 - \sqrt{n})$ and $A(n, n/2 - 2 \sqrt{n})$ are obtained, and, to the best of the authors' knowledge, are the first upper bounds scaling polynomially in n for the regime with $d = n/2 - \Theta(\sqrt{n})$.

A novel permuted fast successive-cancellation list decoding algorithm with fast Hadamard transform (FHT-FSCL) is presented. The proposed decoder initializes $L$ $(L\ge1)$ active decoding paths with $L$ random codeword permutations sampled from the full symmetry group of the codes. The path extension in the permutation domain is carried out until the first constituent RM code of order $1$ is visited. Conventional path extension of the successive-cancellation list decoder is then utilized in the information bit domain. The simulation results show that for a RM code of length $512$ with $46$ information bits, by running $20$ parallel permuted FHT-FSCL decoders with $L=4$, we reduce $72\%$ of the computational complexity, $22\%$ of the decoding latency, and $84\%$ of the memory consumption of the state-of-the-art simplified successive-cancellation decoder that uses $512$ permutations sampled from the full symmetry group of the code, with similar error-correction performance at the target frame error rate of $10^{-4}$.

Locally repairable codes (LRCs) with $(r,\delta)$ locality were introduced by Prakash \emph{et al.} into distributed storage systems (DSSs) due to their benefit of locally repairing at least $\delta-1$ erasures via other $r$ survival nodes among the same local group. An LRC achieving the $(r,\delta)$ Singleton-type bound is called an optimal $(r,\delta)$ LRC. Constructions of optimal $(r,\delta)$ LRCs with longer code length and determining the maximal code length have been an important research direction in coding theory in recent years. In this paper, we conduct further research on the improvement of maximum code length of optimal $(r,\delta)$ LRCs. For $2\delta+1\leq d\leq 2\delta+2$, our upper bounds largely improve the ones by Cai \emph{et al.}, which are tight in some special cases. Moreover, we generalize the results of Chen \emph{et al.} and obtain a complete characterization of optimal $(r=2, \delta)$-LRCs in the sense of geometrical existence in the finite projective plane $PG(2,q)$. Within this geometrical characterization, we construct a class of optimal $(r,\delta)$ LRCs based on the sunflower structure. Both the construction and upper bounds are better than previous ones.

Recently, an interest in constructing pseudorandom or hitting set generators for restricted branching programs has increased, which is motivated by the fundamental issue of derandomizing space-bounded computations. Such constructions have been known only in the case of width 2 and in very restricted cases of bounded width. In this paper, we characterize the hitting sets for read-once branching programs of width 3 by a so-called richness condition. Namely, we show that such sets hit the class of read-once conjunctions of DNF and CNF (i.e. the weak richness). Moreover, we prove that any rich set extended with all strings within Hamming distance of 3 is a hitting set for read-once branching programs of width 3. Then, we show that any almost $O(\log n)$-wise independent set satisfies the richness condition. By using such a set due to Alon et al. (1992) our result provides an explicit polynomial-time construction of a hitting set for read-once branching programs of width 3 with acceptance probability $\varepsilon>5/6$. We announced this result at conferences more than ten years ago, including only proof sketches, which motivated a number of subsequent results on pseudorandom generators for restricted read-once branching programs. This paper contains our original detailed proof that has not been published yet.

This paper presents encoding and decoding algorithms for several families of optimal rank metric codes whose codes are in restricted forms of symmetric, alternating and Hermitian matrices. First, we show the evaluation encoding is the right choice for these codes and then we provide easily reversible encoding methods for each family. Later unique decoding algorithms for the codes are described. The decoding algorithms are interpolation-based and can uniquely correct errors for each code with rank up to $\lfloor(d-1)/2\rfloor$ in polynomial-time, where $d$ is the minimum distance of the code.

We study (Galois) linear complementary dual codes over mixed alphabets arising from finite chain rings. We give a characterization of when a given code is of We study (Galois) linear complementary dual codes over mixed alphabets arising from finite chain rings. We give a characterization of when a given code is of this type and when it is Galois invariant. Finally, this leads to a study of the Gray image of $\mathbb{F}_p\mathbb{F}_p[\theta]$-linear codes, where $p\in\{2; 3\}$ and $\theta\neq\theta^2=0$, that provides $\mathbb{F}_p$-linear complementary dual codes.

In this paper, we are interested in the performance of a variable-length stop-feedback (VLSF) code with $m$ optimal decoding times for the binary-input additive white Gaussian noise channel. We first develop tight approximations on the tail probability of length-$n$ cumulative information density. Building on the work of Yavas \emph{et al.}, for a given information density threshold, we formulate the integer program of minimizing the upper bound on average blocklength over all decoding times subject to the average error probability, minimum gap and integer constraints. Eventually, minimization of locally minimum upper bounds over all thresholds will yield the globally minimum upper bound and this is called the two-step minimization. For the integer program, we present a greedy algorithm that yields possibly suboptimal integer decoding times. By allowing a positive real-valued decoding time, we develop the gap-constrained sequential differential optimization (SDO) procedure that sequentially produces the optimal, real-valued decoding times. We identify the error regime in which Polyanskiy's scheme of stopping at zero does not improve the achievability bound. In this error regime, the two-step minimization with the gap-constrained SDO shows that a finite $m$ suffices to attain Polyanskiy's bound for VLSF codes with $m = \infty$.

Particle smoothers are SMC (Sequential Monte Carlo) algorithms designed to approximate the joint distribution of the states given observations from a state-space model. We propose dSMC (de-Sequentialized Monte Carlo), a new particle smoother that is able to process $T$ observations in $\mathcal{O}(\log T)$ time on parallel architecture. This compares favourably with standard particle smoothers, the complexity of which is linear in $T$. We derive $\mathcal{L}_p$ convergence results for dSMC, with an explicit upper bound, polynomial in $T$. We then discuss how to reduce the variance of the smoothing estimates computed by dSMC by (i) designing good proposal distributions for sampling the particles at the initialization of the algorithm, as well as by (ii) using lazy resampling to increase the number of particles used in dSMC. Finally, we design a particle Gibbs sampler based on dSMC, which is able to perform parameter inference in a state-space model at a $\mathcal{O}(\log(T))$ cost on parallel hardware.