# Codes for Updating Linear Functions over Small Fields

## TL;DR

This paper studies efficient linear coding schemes for updating linear functions of sparse message changes over small finite fields, with applications to distributed data storage, providing tighter bounds and constructions that reduce field size requirements.

## Contribution

It offers a field-size aware analysis of the function update problem, tighter codelength bounds, and new codes that balance codelength reduction with smaller field sizes.

## Key findings

- Derived a tighter lower bound on codelength independent of field size
- Constructed codes for striped message vectors with reduced field size requirements
- Established equivalence between function update and generalized index coding problems

## Abstract

We consider a point-to-point communication scenario where the receiver maintains a specific linear function of a message vector over a finite field. When the value of the message vector undergoes a sparse update, the transmitter broadcasts a coded version of the modified message while the receiver uses this codeword and the current value of the linear function to update its contents. It is assumed that the transmitter has access to the modified message but is unaware of the exact difference vector between the original and modified messages. Under the assumption that the difference vector is sparse and that its Hamming weight is at the most a known constant, the objective is to design a linear code with as small a codelength as possible that allows successful update of the linear function at the receiver. This problem is motivated by applications to distributed data storage systems. Recently, Prakash and Medard derived a lower bound on the codelength, which is independent of the size of the underlying finite field, and provided constructions that achieve this bound if the size of the finite field is sufficiently large. However, this requirement on the field size can be prohibitive for even moderate values of the system parameters. In this paper, we provide a field-size aware analysis of the function update problem, including a tighter lower bound on the codelength, and design codes that trade-off the codelength for a smaller field size requirement. We first characterize the family of function update problems where linear coding can provide reduction in codelength compared to a naive transmission scheme. We then provide field-size dependent bounds on the optimal codelength, and construct coding schemes when the receiver maintains linear functions of striped message vector. Finally, we show that every function update problem is equivalent to a generalized index coding problem.