# $b$-symbol distance distribution of repeated-root cyclic codes

## TL;DR

This paper develops a method to compute the $b$-symbol-pair distance of sequences and analyzes the $b$-symbol-pair distances of certain cyclic codes over finite fields, relevant for high-density data storage channels.

## Contribution

It introduces a new method for calculating $b$-symbol-pair distances and applies it to specific cyclic codes of length $p^e$ over finite fields, expanding coding theory for $b$-symbol read channels.

## Key findings

- Computed $b$-symbol-pair distances for specific cyclic codes
- Provided formulas for $b$-symbol-pair distances of repeated-root cyclic codes
- Enhanced understanding of code performance in $b$-symbol read channels

## Abstract

Symbol-pair codes, introduced by Cassuto and Blaum [1], have been raised for symbol-pair read channels. This new idea is motivated by the limitation of the reading process in high-density data storage technologies. Yaakobi et al. [8] introduced codes for $b$-symbol read channels, where the read operation is performed as a consecutive sequence of $b>2$ symbols. In this paper, we come up with a method to compute the $b$-symbol-pair distance of two $n$-tuples, where $n$ is a positive integer. Also, we deal with the $b$-symbol-pair distances of some kind of cyclic codes of length $p^e$ over $\mathbb{F}_{p^m}$.