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Small-distance block code

Description

A block code of length \(n\) that either detects or corrects errors on at most two coordinates, i.e., has distance \(d \leq 5\).

See Ref. [1] for a comparison of short block codes.

Cousins

Small-distance block quantum code
Nordstrom-Robinson (NR) code— The NR code can be shortened to produce unique \((15, 256, 5)\), \((14, 128, 5)\), and \((13, 64, 5)\) codes [2; pg. 74].

Member of code lists

Classical codes
Small-distance classical codes
Small-distance quantum codes and friends

Primary Hierarchy

Classical Domain

Error-correcting code (ECC)

Parents

Small-distance block code

Children

\([7,3,4]\) simplex code

\([2^r,2^r-r-1,4]\) Extended Hamming code

Petersen cycle code

Best \((10,40,4)\) code

Sloane-Whitehead code

\(q\)-ary Hamming code

\(q\)-ary parity-check code

Ternary Golay code

References

[1]: M. C. Coşkun, G. Durisi, T. Jerkovits, G. Liva, W. Ryan, B. Stein, and F. Steiner, “Efficient Error-Correcting Codes in the Short Blocklength Regime”, (2019) arXiv:1812.08562
[2]: F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.

Page edit log

Victor V. Albert (2023-11-22) — most recent

Cite as:

“Small-distance block code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/small_distance

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/properties/block/small_distance.yml.

Block
ECC
Classical-quantum (c-q)
Melas
\([7,3,4]\) simplex
\([2^r,2^r-r-1,4]\) Extended Hamming
Petersen cycle
Small-distance block quantum
NR

Classical Domain

Quantum Domain

Classical-quantum Domain

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Error correction zoo by Victor V. Albert, Philippe Faist, and many contributors. This work is licensed under a CC-BY-SA License. See how to contribute.