## Description

## Protection

Since they are designed to protect against corruption of nodes in a distributed storage network, many array codes are analyzed in terms of their erasure capabilities.

An analogue of the Hamming metric for array codes comes from the Hamming distance between two codewords whose columns of length \(m\) are converted into values in \(GF(q)^m\) by treating the latter as the \(m\)th extension of \(GF(q)\). This converts a matrix codeword over \(GF(q)\) into a vector over \(GF(q)^m\) and is called subpacketization. Linear array codes are those codes that are linear w.r.t. \(GF(q)^m\), i.e., that are closed under addition and multiplication by elements of said field when in vector form.

There are other notions of distance for array codes, including the rank metric and its generalization the sum-rank metric.

## Notes

## Parents

## Children

- Generalized EVENODD code
- MDS array code
- Regenerating code (RGC)
- Cross-interleaved RS (CIRS) code — The CIRS code can also be visualized as a 2D array code [1].

## Cousin

- Reed-Solomon (RS) code — RS codes over \(q=2^m\) are used in RAID 6 [3,4]; see [1].

## References

- [1]
- M. Blaum, P. G. Farrell, H. C. A. van Tilborg, 1998. Array codes. Handbook of coding theory, 2 (Part 2), pp. 1855-1909.
- [2]
- E. Fujiwara, Code Design for Dependable Systems (Wiley, 2005) DOI
- [3]
- Anvin, H. Peter. "The mathematics of RAID-6." (2007).
- [4]
- S. T. Position. (2009) Common raid disk data format specification. [Online]. Available: http://www.snia.org/tech activities/standards/curr standards/ddf

## Page edit log

- Victor V. Albert (2023-05-04) — most recent

## Cite as:

“Array code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/array

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/matrices/raid/array.yml.