# A Generic Transformation for Optimal Node Repair in MDS Array Codes over $\mathbf{F}_2$

## TL;DR

This paper introduces a new generic transformation for binary MDS array codes that enables optimal repair bandwidth and rebuilding access for all nodes, addressing limitations of previous transformations that did not work for binary codes.

## Contribution

A novel transformation method for binary MDS array codes that ensures optimal repair properties for all nodes while preserving existing efficiencies.

## Key findings

- Transformation applies to any [n,k] binary MDS array code.
- Multiple applications yield codes with optimal rebuilding access for all nodes.
- Existing codes with partial optimality can be extended to full optimality.

## Abstract

For high-rate linear systematic maximum distance separable (MDS) codes, most early constructions could initially optimally repair all the systematic nodes but not all the parity nodes. Fortunately, this issue was first solved by Li et al. in (IEEE Trans. Inform. Theory, 64(9), 6257-6267, 2018), where a transformation that can convert any nonbinary MDS array code into another one with desired properties was proposed. However, the transformation does not work for binary MDS array codes. In this paper, we address this issue by proposing another generic transformation that can convert any $[n, k]$ binary MDS array code into a new one, which endows any $r=n-k\ge2$ chosen nodes with optimal repair bandwidth and optimal rebuilding access properties, and at the same time, preserves the normalized repair bandwidth/rebuilding access for the remaining $k$ nodes under some conditions. As two immediate applications, we show that 1) by applying the transformation multiple times, any binary MDS array code can be converted into one with optimal rebuilding access for all nodes, 2) any binary MDS array code with optimal repair bandwidth or optimal rebuilding access for the systematic nodes can be converted into one with the corresponding optimality property for all nodes.