# An improvement of the asymptotic Elias bound for non-binary codes

## TL;DR

This paper develops a hybrid bound combining Elias and Plotkin bounds for non-binary codes, improving the upper bounds on the asymptotic information rate across different minimum distances.

## Contribution

It introduces a new hybrid bound based on the anticode bound that outperforms existing bounds for non-binary codes.

## Key findings

- The hybrid bound improves upon Elias and Plotkin bounds.
- The anticode bound enhances the understanding of code rate limits.
- Assuming a conjecture on convexity, the bounds follow directly.

## Abstract

For non-binary codes the Elias bound is a good upper bound for the asymptotic information rate at low relative minimum distance, where as the Plotkin bound is better at high relative minimum distance. In this work, we obtain a hybrid of these bounds which improves both. This in turn is based on the anticode bound which is a hybrid of the Hamming and Singleton bounds and improves both bounds. The question of convexity of the asymptotic rate function is an important open question. We conjecture a much weaker form of the convexity, and we show that our bounds follow immediately if we assume the conjecture.