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IEEE Information Theory Workshop (ITW)

On Function-Correcting Codes

Rohit Premlal, Balaji Sundar Rajan

Published Nov 24, 2024• 4 citations
Publisher

Abstract

A class of codes designed to protect function evaluations of a message from errors was introduced in “Function-Correcting Codes” by Lenz et al. 2023. They provide a graphical representation for the problem of constructing functioncorrecting codes. We use this graph to get a lower bound on the redundancy required for function correction and classical error correction. For linear functions, we show that the adjacency matrix of this graph is diagonalised by tensor powers of Discrete Fourier Transform (DFT) matrices, which leads to a lower bound on redundancy. Also, we propose a version of the sphere packing bound for linear-function correcting codes. Further more, we identify a class of linear functions for which an upper bound proposed by Lenz et al., is tight.

DOI
10.1109/itw61385.2024.10807007

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A class of codes designed to protect function evaluations of a message from errors was introduced in “Function-Correcting Codes” by Lenz et al. 2023. They provide a graphical representation for the problem of constructing functioncorrecting codes. We use this graph to get a lower bound on the redundancy required for function correction and classical error correction. For linear functions, we show that the adjacency matrix of this graph is diagonalised by tensor powers of Discrete Fourier Transform (DFT) matrices, which leads to a lower bound on redundancy. Also, we propose a version of the sphere packing bound for linear-function correcting codes. Further more, we identify a class of linear functions for which an upper bound proposed by Lenz et al., is tight.
One of the most important open problems in the communication society is to determine the performance of iterative message passing algorithms over loopy graphs.Some recent work addresses this problem for loopy Tanner graphs by introducing the concepts of stopping distance and stopping redundancy.By analyzing the eigenvalues and eigenvectors of the normalized incidence matrix representing a Tanner graph, we derive lower bounds on its stopping distance.Using these lower bounds, an upper bound on stopping redundancy of the difference-set codes is derived as well.
Upper and lower bounds on the error probability of linear codes under maximum-likelihood (ML) decoding are shortly surveyed and applied to ensembles of codes on graphs. For upper bounds, focus is put on Gallager bounding techniques and their relation to a variety of other reported bounds. Within the class of lower bounds, we address de Caen's based bounds and their improvements, sphere-packing bounds, and information-theoretic bounds on the bit error probability of codes defined on graphs. A comprehensive overview is provided in a monograph by the authors which is currently in preparation.
Ever since the publication of Shannon's seminal work in 1948, the search for capacity achieving codes has led to many interesting discoveries in channel coding theory. Low-density parity-check (LDPC) codes originally proposed in 1963 were largely forgotten and rediscovered recently. The significance of LDPC codes lies in their capacity approaching performance even when decoded using low complexity sub-optimal decoding algorithms. Iterative decoders are one such class of decoders that work on a graphical representation of a code known as the Tanner graph. Their properties have been well understood in the asymptotic limit of the code length going to infinity. However, the behavior of various decoders for a given finite length code remains largely unknown.An understanding of the failures of the decoders is vital for the error floor analysis of a given code. Broadly speaking, error floor is the abrupt degradation in the frame error rate (FER) performance of a code in the high signal-to-noise ratio domain. Since the error floor phenomenon manifests in the regions not reachable by Monte-Carlo simulations, analytical methods are necessary for characterizing the decoding failures. In this work, we consider hard decision decoders for transmission over the binary symmetric channel (BSC).For column-weight-three codes, we provide tight upper and lower bounds on the guaranteed error correction capability of a code under the Gallager A algorithm by studying combinatorial objects known as trapping sets. For higher column weight codes, we establish bounds on the minimum number of variable nodes that achieve certain expansion as a function of the girth of the underlying Tanner graph, thereby obtaining lower bounds on the guaranteed error correction capability. We explore the relationship between a class of graphs known as cage graphs and trapping sets to establish upper bounds on the error correction capability.We also propose an algorithm to identify the most probable noise configurations, also known as instantons, that lead to error floor for linear programming (LP) decoding over the BSC. With the insight gained from the above analysis techniques, we propose novel code construction techniques that result in codes with superior error floor performance.
A fundamental quantity of interest in Shannon theory, classical or quantum, is the error exponent of a given channel W and rate R: the constantE(W,R)E(W,R)E(W,R)which governs the exponential decay of decoding error when using ever larger optimal codes of fixed rate R to communicate over ever more (memoryless) instances of a given channel W. Nearly matching lower and upper bounds are well-known for classical channels. Here I show a lower bound on the error exponent of communication over arbitrary classical-quantum (CQ) channels which matches Dalai’s sphere-packing upper bound for rates above a critical value, exactly analogous to the case of classical channels. This proves a conjecture made by Holevo in his investigation of the problem. Unlike the classical case, however, the argument does not proceed via a refined analysis of a suitable decoder, but instead by leveraging a bound by Hayashi on the error exponent of the cryptographic task of privacy amplification. This bound is then related to the coding problem via tight entropic uncertainty relations and Gallager’s method of constructing capacity-achieving parity-check codes for arbitrary channels. Along the way, I find a lower bound on the error exponent of the task of compression of classical information relative to quantum side information that matches the sphere-packing upper bound of Cheng et al. In turn, the polynomial prefactors to the sphere-packing bound found by Cheng et al. may be translated to the privacy amplification problem, sharpening a recent result by Li, Yao, and Hayashi, at least for linear randomness extractors.