arXiv subjects

Arman Fazeli

Publications and source records attributed to Arman Fazeli.

12 recordsLinked to original sources

arXiv2026

A Reinforcement Learning Based Universal Sequence Design for Polar Codes

To advance Polar code design for 6G applications, we develop a reinforcement learning-based universal sequence design framework that is extensible and adaptable to diverse channel conditions and decoding strategies. Crucially, our method scales to code lengths up to $2048$, making it suitable for use in standardization. Across all $(N,K)$ configurations supported in 5G, our approach achieves competitive performance relative to the NR sequence adopted in 5G and yields up to a 0.2 dB gain over the beta-expansion baseline at $N=2048$. We further highlight the key elements that enabled learning at scale: (i) incorporation of physical law constrained learning grounded in the universal partial order property of Polar codes, (ii) exploitation of the weak long term influence of decisions to limit lookahead evaluation, and (iii) joint multi-configuration optimization to increase learning efficiency.

cs.LG↗

arXiv2026

Partially Polarized Polar Codes: A New Design for 6G Control Channels

We introduce a new family of polar-like codes, called Partially Polarized Polar (PPP) codes. PPP codes are constructed from conventional polar codes by selectively pruning polarization kernels, thereby modifying the synthesized bit-channel capacities to ensure a guaranteed number of non-frozen bits available early in decoding. These early-access information bits enable more effective early termination, which is particularly valuable for blind decoding in downlink control channels, where user equipment (UE) must process multiple candidates, many of which carry no valid control information. Our results show that PPP codes offer substantial performance gains over conventional polar codes, particularly at larger block lengths where hardware limitations restrict straightforward scaling. Compared with existing methods such as aggregation or segmentation, PPP codes achieve higher efficiency without the need for additional hardware support. Finally, we propose several frozen-bitmap design strategies tailored to PPP codes.

cs.IT↗

arXiv2021

A Deterministic Algorithm for Computing the Weight Distribution of Polar Codes

In this work, we present a deterministic algorithm for computing the entire weight distribution of polar codes. As the first step, we derive an efficient recursive procedure to compute the weight distribution that arises in successive cancellation decoding of polar codes along any decoding path. This solves the open problem recently posed by Polyanskaya, Davletshin, and Polyanskii. Using this recursive procedure, at code length n, we can compute the weight distribution of any polar cosets in time O(n^2). We show that any polar code can be represented as a disjoint union of such polar cosets; moreover, this representation extends to polar codes with dynamically frozen bits. However, the number of polar cosets in such representation scales exponentially with a parameter introduced herein, which we call the mixing factor. To upper bound the complexity of our algorithm for polar codes being decreasing monomial codes, we study the range of their mixing factors. We prove that among all decreasing monomial codes with rates at most 1/2, self-dual Reed-Muller codes have the largest mixing factors. To further reduce the complexity of our algorithm, we make use of the fact that, as decreasing monomial codes, polar codes have a large automorphism group. That automorphism group includes the block lower-triangular affine group (BLTA), which in turn contains the lower-triangular affine group (LTA). We prove that a subgroup of LTA acts transitively on certain subsets of decreasing monomial codes, thereby drastically reducing the number of polar cosets that we need to evaluate. This complexity reduction makes it possible to compute the weight distribution of polar codes at length n = 128.

cs.IT↗

arXiv2020

Parallelism versus Latency in Simplified Successive-Cancellation Decoding of Polar Codes

This paper characterizes the latency of the simplified successive-cancellation (SSC) decoding scheme for polar codes under hardware resource constraints. In particular, when the number of processing elements $P$ that can perform SSC decoding operations in parallel is limited, as is the case in practice, the latency of SSC decoding is $O\left(N^{1-1/μ}+\frac{N}{P}\log_2\log_2\frac{N}{P}\right)$, where $N$ is the block length of the code and $μ$ is the scaling exponent of the channel. Three direct consequences of this bound are presented. First, in a fully-parallel implementation where $P=\frac{N}{2}$, the latency of SSC decoding is $O\left(N^{1-1/μ}\right)$, which is sublinear in the block length. This recovers a result from our earlier work. Second, in a fully-serial implementation where $P=1$, the latency of SSC decoding scales as $O\left(N\log_2\log_2 N\right)$. The multiplicative constant is also calculated: we show that the latency of SSC decoding when $P=1$ is given by $\left(2+o(1)\right) N\log_2\log_2 N$. Third, in a semi-parallel implementation, the smallest $P$ that gives the same latency as that of the fully-parallel implementation is $P=N^{1/μ}$. The tightness of our bound on SSC decoding latency and the applicability of the foregoing results is validated through extensive simulations.

cs.IT↗

arXiv2020

Binary Linear Codes with Optimal Scaling: Polar Codes with Large Kernels

We prove that, for the binary erasure channel (BEC), the polar-coding paradigm gives rise to codes that not only approach the Shannon limit but do so under the best possible scaling of their block length as a~function of the gap to capacity. This result exhibits the first known family of binary codes that attain both optimal scaling and quasi-linear complexity of encoding and decoding. Our proof is based on the construction and analysis of binary polar codes with large kernels. When communicating reliably at rates within $\varepsilon > 0$ of capacity, the code length $n$ often scales as $O(1/\varepsilon^μ)$, where the constant $μ$ is called the scaling exponent. It is known that the optimal scaling exponent is $μ=2$, and it is achieved by random linear codes. The scaling exponent of conventional polar codes (based on the $2\times 2$ kernel) on the BEC is $μ=3.63$. This falls far short of the optimal scaling guaranteed by random codes. Our main contribution is a rigorous proof of the following result: for the BEC, there exist $\ell\times\ell$ binary kernels, such that polar codes constructed from these kernels achieve scaling exponent $μ(\ell)$ that tends to the optimal value of $2$ as $\ell$ grows. We furthermore characterize precisely how large $\ell$ needs to be as a function of the gap between $μ(\ell)$ and $2$. The resulting binary codes maintain the recursive structure of conventional polar codes, and thereby achieve construction complexity $O(n)$ and encoding/decoding complexity $O(n\log n)$.

cs.IT↗

arXiv2020

Polar Codes for the Deletion Channel: Weak and Strong Polarization

This paper presents the first proof of polarization for the deletion channel with a constant deletion rate and a regular hidden-Markov input distribution. A key part of this work involves representing the deletion channel using a trellis and describing the plus and minus polar-decoding operations on that trellis. In particular, the plus and minus operations can be seen as combining adjacent trellis stages to yield a new trellis with half as many stages. Using this viewpoint, we prove a weak polarization theorem for standard polar codes on the deletion channel. To achieve strong polarization, we modify this scheme by adding guard bands of repeated zeros between various parts of the codeword. This gives a scheme whose rate approaches the mutual information and whose probability of error decays exponentially in the cube-root of the block length. We conclude by showing that this scheme can achieve capacity on the deletion channel by proving that the capacity of the deletion channel can be achieved by a sequence of regular hidden-Markov input distributions.

cs.IT↗

arXiv2020

List Decoding of Arikan's PAC Codes

Polar coding gives rise to the first explicit family of codes that provably achieve capacity with efficient encoding and decoding for a wide range of channels. However, its performance at short block lengths is far from optimal. Arikan has recently presented a new polar coding scheme, which he called polarization-adjusted convolutional (PAC) codes. Such PAC codes provide dramatic improvement in performance as compared to both standard successive-cancellation decoding as well as CRC-aided list decoding. Arikan's PAC codes are based primarily upon the following ideas: replacing CRC precoding with convolutional precoding (under appropriate rate profiling) and replacing list decoding by sequential decoding. His simulations show that PAC codes, resulting from the combination of these ideas, are close to finite-length bounds on the performance of any code under ML decoding. One of our main goals in this paper is to answer the following question: is sequential decoding essential for the superior performance of PAC codes? We show that similar performance can be achieved using list decoding when the list size $L$ is moderately large (say, $L \ge 128$). List decoding has distinct advantages over sequential decoding is certain scenarios, such as low-SNR regimes or situations where the worst-case complexity/latency is the primary constraint. Another objective is to provide some insights into the remarkable performance of PAC codes. We first observe that both sequential decoding and list decoding of PAC codes closely match ML decoding thereof. We then estimate the number of low weight codewords in PAC codes, using these estimates to approximate the union bound on their performance under ML decoding. These results indicate that PAC codes are superior to both polar codes and Reed-Muller codes, and suggest that the goal of rate-profiling may be to optimize the weight distribution at low weights.

cs.IT↗

arXiv2019

Explicit Polar Codes with Small Scaling Exponent

Herein, we focus on explicit constructions of $\ell\times\ell$ binary kernels with small scaling exponent for $\ell \le 64$. In particular, we exhibit a sequence of binary linear codes that approaches capacity on the BEC with quasi-linear complexity and scaling exponent $μ< 3$. To the best of our knowledge, such a sequence of codes was not previously known to exist. The principal challenges in establishing our results are twofold: how to construct such kernels and how to evaluate their scaling exponent. In a single polarization step, an $\ell\times\ell$ kernel $K_\ell$ transforms an underlying BEC into $\ell$ bit-channels $W_1,W_2,\ldots,W_\ell$. The erasure probabilities of $W_1,W_2,\ldots,W_\ell$, known as the polarization behavior of $K_\ell$, determine the resulting scaling exponent $μ(K_\ell)$. We first introduce a class of self-dual binary kernels and prove that their polarization behavior satisfies a strong symmetry property. This reduces the problem of constructing $K_\ell$ to that of producing a certain nested chain of only $\ell/2$ self-orthogonal codes. We use nested cyclic codes, whose distance is as high as possible subject to the orthogonality constraint, to construct the kernels $K_{32}$ and $K_{64}$. In order to evaluate the polarization behavior of $K_{32}$ and $K_{64}$, two alternative trellis representations (which may be of independent interest) are proposed. Using the resulting trellises, we show that $μ(K_{32})=3.122$ and explicitly compute over half of the polarization behavior coefficients for $K_{64}$, at which point the complexity becomes prohibitive. To complete the computation, we introduce a Monte-Carlo interpolation method, which produces the estimate $μ(K_{64})\simeq 2.87$. We augment this estimate with a rigorous proof that $μ(K_{64})<2.97$.

cs.IT↗

arXiv2016

Minimum Storage Regenerating Codes For All Parameters

Regenerating codes for distributed storage have attracted much research interest in the past decade. Such codes trade the bandwidth needed to repair a failed node with the overall amount of data stored in the network. Minimum storage regenerating (MSR) codes are an important class of optimal regenerating codes that minimize (first) the amount of data stored per node and (then) the repair bandwidth. Specifically, an $[n,k,d]$-$(α)$ MSR code $\mathbb{C}$ over $\mathbb{F}_q$ is defined as follows. Using such a code $\mathbb{C}$, a file $\cal{F}$ consisting of $αk$ symbols over $\mathbb{F}_q$ can be distributed among $n$ nodes, each storing $α$ symbols, in such a way that: The file $\cal{F}$ can be recovered by downloading the content of any $k$ of the $n$ nodes; and the content of any failed node can be reconstructed by accessing any $d$ of the remaining $n-1$ nodes and downloading $α/(d-k+1)$ symbols from each of these nodes. Unfortunately, explicit constructions of $[n,k,d]$ MSR codes are known only for certain special cases: either low rate, namely $k/n<0.5$, or high repair connectivity, namely $d = n-1$. Although setting $d = n-1$ minimizes the repair bandwidth, it may be impractical to connect to all the remaining nodes in order to repair a single failed node. Our main result in this paper is an explicit construction of systematic-repair $[n,k,d]$ MSR codes for all possible values of parameters $n,k,d$. In particular, we construct systematic-repair MSR codes of high rate $k/n>0.5$ and low repair connectivity $k< d<n-1$. Such codes were not previously known to exist. In order to construct these codes, we solve simultaneously several repair scenarios, each of which is expressible as an interference alignment problem. Extension of our results beyond systematic repair remains an open problem.

cs.IT↗

arXiv2015

PIR with Low Storage Overhead: Coding instead of Replication

Private information retrieval (PIR) protocols allow a user to retrieve a data item from a database without revealing any information about the identity of the item being retrieved. Specifically, in information-theoretic $k$-server PIR, the database is replicated among $k$ non-communicating servers, and each server learns nothing about the item retrieved by the user. The cost of PIR protocols is usually measured in terms of their communication complexity, which is the total number of bits exchanged between the user and the servers, and storage overhead, which is the ratio between the total number of bits stored on all the servers and the number of bits in the database. Since single-server information-theoretic PIR is impossible, the storage overhead of all existing PIR protocols is at least $2$. In this work, we show that information-theoretic PIR can be achieved with storage overhead arbitrarily close to the optimal value of $1$, without sacrificing the communication complexity. Specifically, we prove that all known $k$-server PIR protocols can be efficiently emulated, while preserving both privacy and communication complexity but significantly reducing the storage overhead. To this end, we distribute the $n$ bits of the database among $s+r$ servers, each storing $n/s$ coded bits (rather than replicas). For every fixed $k$, the resulting storage overhead $(s+r)/s$ approaches $1$ as $s$ grows; explicitly we have $r\le k\sqrt{s}(1+o(1))$. Moreover, in the special case $k = 2$, the storage overhead is only $1 + \frac{1}{s}$. In order to achieve these results, we introduce and study a new kind of binary linear codes, called here $k$-server PIR codes. We then show how such codes can be constructed, and we establish several bounds on the parameters of $k$-server PIR codes. Finally, we briefly discuss extensions of our results to nonbinary alphabets, to robust PIR, and to $t$-private PIR.

cs.IT↗

arXiv2014

Generalized Sphere Packing Bound

Kulkarni and Kiyavash recently introduced a new method to establish upper bounds on the size of deletion-correcting codes. This method is based upon tools from hypergraph theory. The deletion channel is represented by a hypergraph whose edges are the deletion balls (or spheres), so that a deletion-correcting code becomes a matching in this hypergraph. Consequently, a bound on the size of such a code can be obtained from bounds on the matching number of a hypergraph. Classical results in hypergraph theory are then invoked to compute an upper bound on the matching number as a solution to a linear-programming problem. The method by Kulkarni and Kiyavash can be applied not only for the deletion channel but also for other error channels. This paper studies this method in its most general setup. First, it is shown that if the error channel is regular and symmetric then this upper bound coincides with the sphere packing bound and thus is called the generalized sphere packing bound. Even though this bound is explicitly given by a linear programming problem, finding its exact value may still be a challenging task. In order to simplify the complexity of the problem, we present a technique based upon graph automorphisms that in many cases reduces the number of variables and constraints in the problem. We then apply this method on specific examples of error channels. We start with the $Z$ channel and show how to exactly find the generalized sphere packing bound for this setup. Next studied is the non-binary limited magnitude channel both for symmetric and asymmetric errors, where we focus on the single-error case. We follow up on the deletion and grain-error channels and show how to improve upon the existing upper bounds for single deletion/error. Finally, we apply this method for projective spaces and find its generalized sphere packing bound for the single-error case.

cs.IT↗

arXiv2013

Nontrivial t-Designs over Finite Fields Exist for All t

A $t$-$(n,k,λ)$ design over $\F_q$ is a collection of $k$-dimensional subspaces of $\F_q^n$, called blocks, such that each $t$-dimensional subspace of $\F_q^n$ is contained in exactly $λ$ blocks. Such $t$-designs over $\F_q$ are the $q$-analogs of conventional combinatorial designs. Nontrivial $t$-$(n,k,λ)$ designs over $\F_q$ are currently known to exist only for $t \leq 3$. Herein, we prove that simple (meaning, without repeated blocks) nontrivial $t$-$(n,k,λ)$ designs over $\F_q$ exist for all $t$ and $q$, provided that $k > 12t$ and $n$ is sufficiently large. This may be regarded as a $q$-analog of the celebrated Teirlinck theorem for combinatorial designs.

math.CO↗