On the Height Profile of Analog Error-Correcting Codes

TL;DR

This paper studies the height profile of analog error-correcting codes, providing new characterizations and computational methods to understand their error detection and correction capabilities beyond traditional minimum distance metrics.

Contribution

It introduces a novel characterization of the height profile, formulates it as a linear programming problem, and offers geometric interpretations for certain code families.

Findings

Characterizations of the height profile are developed.

Methods for computing the height profile via linear programs are presented.

Explicit calculations of the height profile for various code families are demonstrated.

Abstract

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 $\delta < \Delta$, additive errors of magnitude at most $\delta$ are tolerable and need no handling, yet outlying errors of magnitude greater than $\Delta$ are to be located or detected. The trade-off between the ratio $\Delta/\delta$ 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.