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Analog Error Correcting Codes with Constant Redundancy

Wentu Song, Kui Cai

TL;DR

An upper bound on the height profile of analog ECCs whose parity check matrix has columns of unit Euclidean norm is presented and a family of single error-correcting analog ECCs with redundancy three is constructed, which has smaller height profile compared to the known MDS constructions.

Abstract

Analog error-correcting codes (analog ECCs) introduced by Roth are designed to correct outlying errors arising in analog implementations of vector-matrix multiplication. The error-detection/correction capability of an analog ECC can be characterized by its height profile, which is expected to be as small as possible. In this paper, we consider analog ECCs whose parity check matrix has columns of unit Euclidean norm. We first present an upper bound on the height profile of such codes as well as a simple decoder for correcting a single error. We then construct a family of single error-correcting analog ECCs with redundancy three for any code length, which has smaller height profile compared to the known MDS constructions.

Analog Error Correcting Codes with Constant Redundancy

TL;DR

An upper bound on the height profile of analog ECCs whose parity check matrix has columns of unit Euclidean norm is presented and a family of single error-correcting analog ECCs with redundancy three is constructed, which has smaller height profile compared to the known MDS constructions.

Abstract

Analog error-correcting codes (analog ECCs) introduced by Roth are designed to correct outlying errors arising in analog implementations of vector-matrix multiplication. The error-detection/correction capability of an analog ECC can be characterized by its height profile, which is expected to be as small as possible. In this paper, we consider analog ECCs whose parity check matrix has columns of unit Euclidean norm. We first present an upper bound on the height profile of such codes as well as a simple decoder for correcting a single error. We then construct a family of single error-correcting analog ECCs with redundancy three for any code length, which has smaller height profile compared to the known MDS constructions.
Paper Structure (4 sections, 10 theorems, 32 equations, 1 table)

This paper contains 4 sections, 10 theorems, 32 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A class of analog error correcting codes
  4. Single error correcting codes with redundancy three

Key Result

Lemma 1

Roth20 A linear code $\mathcal{C}$ can correct $\tau$ errors and detect $\sigma$ additional errors with respect to the threshold pair $(\delta, \Delta)$ if and only if $2(\mathsf{h}_{2\tau+\sigma}(\mathcal{C})+1)\leq\frac{\Delta}{\delta}$, or equivalently, $\Gamma_{2\tau+\sigma}(\mathcal{C})\leq\fra

Theorems & Definitions (21)

  • Definition 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • proof 1
  • Theorem 1
  • proof 2
  • Remark 1
  • Lemma 4
  • proof 3
  • ...and 11 more