arXiv subjects

Paul H. Siegel

Publications and source records attributed to Paul H. Siegel.

At least 19 recordsLinked to original sources

arXiv2026

A New Class of Geometric Analog Error Correction Codes for Crossbar Based In-Memory Computing

Analog error correction codes have been proposed for analog in-memory computing on resistive crossbars, which can accelerate vector-matrix multiplication for machine learning. Unlike traditional communication or storage channels, this setting involves a mixed noise model with small perturbations and outlier errors. A number of analog codes have been proposed for handling a single outlier, and several constructions have also been developed to address multiple outliers. However, the set of available code families remains limited, covering only a narrow range of code lengths and dimensions. In this paper, we study a recently proposed family of geometric codes capable of handling multiple outliers, and develop a geometric analysis that characterizes their m-height profiles.

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arXiv2026

On the Height Profile of Analog Error-Correcting Codes

In recent work, it has been shown that maintaining reliability in analog vector--matrix multipliers can be modeled as the following coding problem. Vectors in $\mathbb{R}^k$ are encoded into codewords of a linear $[n,k,d]$ code $C$ over $\mathbb{R}$. For prescribed positive reals $δ< Δ$, additive errors of magnitude at most $δ$ are tolerable and need no handling, yet outlying errors of magnitude greater than $Δ$ are to be located or detected. The trade-off between the ratio $Δ/δ$ and the number of outlying errors that can be handled is determined by the height profile of $C$; as such, the height profile provides a finer description of the error handling capability of $C$, compared to the minimum distance $d$, which only determines the number of correctable errors. This work contains a further study of the notion of the height profile. Several characterizations of the height profile are presented, thereby yielding methods for computing it. The starting point is formulating this computation as an optimization problem that is solved by a set of linear programs. This, in turn, leads to a combinatorial characterization of the height profile as a maximum (or max--min) over a certain finite set of codewords of $C$. Moreover, this characterization is shown to have a simple geometric interpretation when the columns of the generator matrix of $C$ all have the same $L_2$ norm. Through examples of several code families, it is demonstrated how the results herein can be used to compute the height profile explicitly.

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arXiv2025

The Labeled Coupon Collector Problem

We generalize the well-known Coupon Collector Problem (CCP) in combinatorics. Our problem is to find the minimum and expected number of draws, with replacement, required to recover $n$ distinctly labeled coupons, with each draw consisting of a random subset of $k$ different coupons and a random ordering of their associated labels. We specify two variations of the problem, Type-I in which the set of labels is known at the start, and Type-II in which the set of labels is unknown at the start. We show that our problem can be viewed as an extension of the separating system problem introduced by Rényi and Katona, provide a full characterization of the minimum, and provide a numerical approach to finding the expectation using a Markov chain model, with special attention given to the case where two coupons are drawn at a time.

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arXiv2025

Multivariate Analytic Combinatorics for Cost Constrained Channels

Analytic combinatorics in several variables is a branch of mathematics that deals with deriving the asymptotic behavior of combinatorial quantities by analyzing multivariate generating functions. We study information-theoretic questions about sequences in a discrete noiseless channel under cost constraints. Our main contributions involve the relationship between the graph structure of the channel and the singularities of the bivariate generating function whose coefficients are the number of sequences satisfying the constraints. We use these new results to invoke theorems from multivariate analytic combinatorics to obtain the asymptotic behavior of the number of cost-limited strings that are admissible by the channel. This builds a new bridge between analytic combinatorics in several variables and labeled weighted graphs, bringing a new perspective and a set of powerful results to the literature of cost-constrained channels. Along the way, we show that the cost-constrained channel capacity is determined by a cost-dependent singularity of the bivariate generating function, generalizing Shannon's classical result for unconstrained capacity, and provide a new proof of the equivalence of the combinatorial and probabilistic definitions of the cost-constrained capacity.

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arXiv2024

Stopping Set Analysis for Concatenated Polar Code Architectures

This paper investigates properties of concatenated polar codes and their potential applications. We start with reviewing previous work on stopping set analysis for conventional polar codes, which we extend in this paper to concatenated architectures. Specifically, we present a stopping set analysis for the factor graph of concatenated polar codes, deriving an upper bound on the size of the minimum stopping set. To achieve this bound, we propose new bounds on the size of the minimum stopping set for conventional polar code factor graphs. The tightness of these proposed bounds is investigated empirically and analytically. We show that, in some special cases, the exact value of the minimum stopping set can be determined with a time complexity of $O(N)$, where $N$ is the codeword length. The stopping set analysis motivates a novel construction method for concatenated polar codes. This method is used to design outer polar codes for two previously proposed concatenated polar code architectures: augmented polar codes and local-global polar codes. Simulation results demonstrate the advantage of the proposed codes over previously proposed constructions based on density evolution (DE).

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arXiv2024

Outer Code Designs for Augmented and Local-Global Polar Code Architectures

In this paper, we introduce two novel methods to design outer polar codes for two previously proposed concatenated polar code architectures: augmented polar codes and local-global polar codes. These methods include a stopping set (SS) construction and a nonstationary density evolution (NDE) construction. Simulation results demonstrate the advantage of these methods over previously proposed constructions based on density evolution (DE) and LLR evolution.

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arXiv2023

A New Version of q-ary Varshamov-Tenengolts Codes with More Efficient Encoders: The Differential VT Codes and The Differential Shifted VT Codes

The problem of correcting deletions and insertions has recently received significantly increased attention due to the DNA-based data storage technology, which suffers from deletions and insertions with extremely high probability. In this work, we study the problem of constructing non-binary burst-deletion/insertion correcting codes. Particularly, for the quaternary alphabet, our designed codes are suited for correcting a burst of deletions/insertions in DNA storage. Non-binary codes correcting a single deletion or insertion were introduced by Tenengolts [1984], and the results were extended to correct a fixed-length burst of deletions or insertions by Schoeny et al. [2017]. Recently, Wang et al. [2021] proposed constructions of non-binary codes of length n, correcting a burst of length at most two for q-ary alphabets with redundancy log n+O(log q log log n) bits, for arbitrary even q. The common idea in those constructions is to convert non-binary sequences into binary sequences, and the error decoding algorithms for the q-ary sequences are mainly based on the success of recovering the corresponding binary sequences, respectively. In this work, we look at a natural solution in which the error detection and correction algorithms are performed directly over q-ary sequences, and for certain cases, our codes provide a more efficient encoder with lower redundancy than the best-known encoder in the literature.

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arXiv2023

Polar Codes with Local-Global Decoding

In this paper, we investigate a coupled polar code architecture that supports both local and global decoding. This local-global construction is motivated by practical applications in data storage and transmission where reduced-latency recovery of sub-blocks of the coded information is required. Local decoding allows random access to sub-blocks of the full code block. When local decoding performance is insufficient, global decoding provides improved data reliability. The coupling scheme incorporates a systematic outer polar code and a partitioned mapping of the outer codeword to semipolarized bit-channels of the inner polar codes. Error rate simulation results are presented for 2 and 4 sub-blocks. Design issues affecting the trade-off between local and global decoding performance are also discussed.

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arXiv2022

Every Bit Counts: A New Version of Non-binary VT Codes with More Efficient Encoder

In this work, we present a new version of non-binary VT codes that are capable of correcting a single deletion or single insertion. Moreover, we provide the first known linear time algorithms that encode user messages into these codes of length n over the $q$-ary alphabet for $q\ge 2$ with at most $\ceil{\log_q n} + 1$ redundant symbols, while the optimal redundancy required is at least $\log_q n + \log_q (q - 1)$ symbols. Our designed encoder reduces the redundancy of the best-known encoder of Tenengolts (1984) by at least $2+\log_q(3)$ redundant symbols, or equivalently $2\log_2 q+3$ redundant bits.

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arXiv2022

Spatio-Temporal Modeling for Flash Memory Channels Using Conditional Generative Nets

We propose a data-driven approach to modeling the spatio-temporal characteristics of NAND flash memory read voltages using conditional generative networks. The learned model reconstructs read voltages from an individual memory cell based on the program levels of the cell and its surrounding cells, as well as the specified program/erase (P/E) cycling time stamp. We evaluate the model over a range of time stamps using the cell read voltage distributions, the cell level error rates, and the relative frequency of errors for patterns most susceptible to inter-cell interference (ICI) effects. We conclude that the model accurately captures the spatial and temporal features of the flash memory channel.

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arXiv2022

Rate-Constrained Shaping Codes for Finite-State Channels With Cost

Shaping codes are used to generate code sequences in which the symbols obey a prescribed probability distribution. They arise naturally in the context of source coding for noiseless channels with unequal symbol costs. Recently, shaping codes have been proposed to extend the lifetime of flash memory and reduce DNA synthesis time. In this paper, we study a general class of shaping codes for noiseless finite-state channels with cost and i.i.d. sources. We establish a relationship between the code rate and minimum average symbol cost. We then determine the rate that minimizes the average cost per source symbol (total cost). An equivalence is established between codes minimizing average symbol cost and codes minimizing total cost, and a separation theorem is proved, showing that optimal shaping can be achieved by a concatenation of optimal compression and optimal shaping for a uniform i.i.d. source.

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arXiv2022

Adaptive Read Thresholds for NAND Flash

A primary source of increased read time on NAND flash comes from the fact that in the presence of noise, the flash medium must be read several times using different read threshold voltages for the decoder to succeed. This paper proposes an algorithm that uses a limited number of re-reads to characterize the noise distribution and recover the stored information. Both hard and soft decoding are considered. For hard decoding, the paper attempts to find a read threshold minimizing bit-error-rate (BER) and derives an expression for the resulting codeword-error-rate. For soft decoding, it shows that minimizing BER and minimizing codeword-error-rate are competing objectives in the presence of a limited number of allowed re-reads, and proposes a trade-off between the two. The proposed method does not require any prior knowledge about the noise distribution, but can take advantage of such information when it is available. Each read threshold is chosen based on the results of previous reads, following an optimal policy derived through a dynamic programming backward recursion. The method and results are studied from the perspective of an SLC Flash memory with Gaussian noise for each level but the paper explains how the method could be extended to other scenarios.

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arXiv2021

Variable-Length Constrained Coding and Kraft Conditions: The Parity-Preserving Case

Previous work by the authors on parity-preserving fixed-length constrained encoders is extended to the variable-length case. Parity-preserving variable-length encoders are formally defined, and, to this end, Kraft conditions are developed for the parity-preserving variable-length setting. Then, a necessary and sufficient condition is presented for the existence of deterministic parity-preserving variable-length encoders for a given constraint. Examples are provided that show that there are coding ratios where parity-preserving variable-length encoders exist, while fixed-length encoders do not.

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arXiv2020

Polar Coding for Multi-level 3-Receiver Broadcast Channels

We consider achieving the rates in the capacity region of a multi-level 3-receiver broadcast channel, in which the second receiver is degraded with respect to the first receiver, with degraded message sets. We propose a two-level chaining strategy based on polar codes that achieves the capacity region of the considered setting without time-sharing. We also look at a slight variation of this problem, where the first receiver only requires to decode its own private message and the other two receivers require to decode another private message common to them. We observe that the capacity region does not enlarge and so the proposed polar coding strategy achieves the capacity region for this problem as well.

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arXiv2020

PR-NN: RNN-based Detection for Coded Partial-Response Channels

In this paper, we investigate the use of recurrent neural network (RNN)-based detection of magnetic recording channels with inter-symbol interference (ISI). We refer to the proposed detection method, which is intended for recording channels with partial-response equalization, as Partial-Response Neural Network (PR-NN). We train bi-directional gated recurrent units (bi-GRUs) to recover the ISI channel inputs from noisy channel output sequences and evaluate the network performance when applied to continuous, streaming data. The computational complexity of PR-NN during the evaluation process is comparable to that of a Viterbi detector. The recording system on which the experiments were conducted uses a rate-2/3, (1,7) runlength-limited (RLL) code with an E2PR4 partial-response channel target. Experimental results with ideal PR signals show that the performance of PR-NN detection approaches that of Viterbi detection in additive white gaussian noise (AWGN). Moreover, the PR-NN detector outperforms Viterbi detection and achieves the performance of Noise-Predictive Maximum Likelihood (NPML) detection in additive colored noise (ACN) at different channel densities. A PR-NN detector trained with both AWGN and ACN maintains the performance observed under separate training. Similarly, when trained with ACN corresponding to two different channel densities, PR-NN maintains its performance at both densities. Experiments confirm that this robustness is consistent over a wide range of signal-to-noise ratios (SNRs). Finally, PR-NN displays robust performance when applied to a more realistic magnetic recording channel with MMSE-equalized Lorentzian signals.

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arXiv2020

On the Performance of Direct Shaping Codes

In this work, we study a recently proposed direct shaping code for flash memory. This rate-1 code is designed to reduce the wear for SLC (one bit per cell) flash by minimizing the average fraction of programmed cells when storing structured data. Then we describe an adaptation of this algorithm that provides data shaping for MLC (two bits per cell) flash memory. It makes use of a page-dependent cost model and is designed to be compatible with the standard procedure of row-by-row, page-based, wordline programming. We also give experimental results demonstrating the performance of MLC data shaping codes when applied to English and Chinese language text. We then study the potential error propagation properties of direct shaping codes when used in a noisy flash device. In particular, we model the error propagation as a biased random walk in a multidimensional space. We prove an upper bound on the error propagation probability and propose an algorithm that can numerically approach a lower bound. Finally, we study the asymptotic performance of direct shaping codes. We prove that the SLC direct shaping code is suboptimal in the sense that it can only achieve the minimum average cost for a rate-1 code under certain conditions on the source distribution.

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arXiv2020

Covering Codes using Insertions or Deletions

A covering code is a set of codewords with the property that the union of balls, suitably defined, around these codewords covers an entire space. Generally, the goal is to find the covering code with the minimum size codebook. While most prior work on covering codes has focused on the Hamming metric, we consider the problem of designing covering codes defined in terms of either insertions or deletions. First, we provide new sphere-covering lower bounds on the minimum possible size of such codes. Then, we provide new existential upper bounds on the size of optimal covering codes for a single insertion or a single deletion that are tight up to a constant factor. Finally, we derive improved upper bounds for covering codes using $R\geq 2$ insertions or deletions. We prove that codes exist with density that is only a factor $O(R \log R)$ larger than the lower bounds for all fixed~$R$. In particular, our upper bounds have an optimal dependence on the word length, and we achieve asymptotic density matching the best known bounds for Hamming distance covering codes.

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arXiv2020

On Bi-Modal Constrained Coding

Bi-modal (respectively, multi-modal) constrained coding refers to an encoding model whereby a user input block can be mapped to two (respectively, multiple) codewords. In current storage applications, such as optical disks, multi-modal coding allows to achieve DC control, in addition to satisfying the runlength limited (RLL) constraint specified by the recording channel. In this work, a study is initiated on bi-modal fixed-length constrained encoders. Necessary and sufficient conditions are presented for the existence of such encoders for a given constraint. It is also shown that under somewhat stronger conditions, one can guarantee a bi-modal encoder with finite decoding delay.

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