Linear Analog Codes: The Good and The Bad

TL;DR

This paper develops a new theoretical framework for linear analog error correction codes, introducing a novel metric called the minimum distance ratio, and demonstrates how to design codes that optimize this metric for minimal mean square error.

Contribution

It introduces the minimum distance ratio metric and the concept of MDRE codes, providing a new approach to optimize analog error correction coding performance.

Findings

Maximum distance ratio correlates with minimal MSE.

Existing codes can be designed to achieve MDRE and MDS simultaneously.

The new metric guides the design of more effective analog codes.

Abstract

This paper studies the theory of linear analog error correction coding. Since classical concepts of minimum Hamming distance and minimum Euclidean distance fail in the analog context, a new metric, termed the "minimum (squared Euclidean) distance ratio," is defined. It is shown that linear analog codes that achieve the largest possible value of minimum distance ratio also achieve the smallest possible mean square error (MSE). Based on this achievability, a concept of "maximum distance ratio expansible (MDRE)" is established, in a spirit similar to maximum distance separable (MDS). Existing codes are evaluated, and it is shown that MDRE and MDS can be simultaneously achieved through careful design.