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\([15,4,8]\) simplex code

Description

Binary code that corrects three bit errors on fifteen bits and generates the stabilizers of the \([[15,7,3]]\) quantum Hamming code. It also generates the \(Z\)-type stabilizers of the \([[15,6,3]]\) and \([[15,4,3]]\) codes. It is the \(m=4\) member of the simplex family, i.e., the dual of the \([15,11,3]\) Hamming code.

Cousins

  • \([[15,4,3]]\) quantum Hamming subcode— The \(Z\)-type stabilizers of the \([[15,4,3]]\) code are generated by the \([15,4,8]\) simplex code.
  • \([[15,6,3]]\) quantum Hamming subcode— The \(Z\)-type stabilizers of the \([[15,6,3]]\) code are generated by the \([15,4,8]\) simplex code, and its \(X\)-type stabilizers by the simplex code together with the all-ones vector.
  • \([[15, 7, 3]]\) quantum Hamming code— The \([[15,7,3]]\) quantum Hamming code is constructed from the \([15,4,8]\) simplex code and its dual \([15,11,3]\) Hamming code via the CSS construction.

Member of code lists

Primary Hierarchy

Classical Domain
Binary Kingdom
Binary codeECC
Binary group-orbit codeECC
Linear binary codeLinear \(q\)-ary Linear code over \(\mathbb{Z}_q\) ECC
Cyclic linear binary codeLinear \(q\)-ary Cyclic Constacyclic Skew-cyclic ECC
Binary BCH codeLinear \(q\)-ary Cyclic Constacyclic Skew-cyclic ECC
\([2^m-1,m,2^{m-1}]\) simplex codeGRM Evaluation Projective geometry Linear \(q\)-ary LCC LRC Distributed-storage AG Universally optimal ECC
Parents
\([2^m-1,m,2^{m-1}]\) simplex code
\([15,4,8]\) simplex code
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Zoo Code ID: simplex_15_4_8

Cite as:
“\([15,4,8]\) simplex code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/simplex_15_4_8, arXiv:2606.11484
BibTeX:
@incollection{eczoo_simplex_15_4_8,
title={\([15,4,8]\) simplex code},
booktitle={The Error Correction Zoo},
year={2026},
editor={Albert, Victor V. and Faist, Philippe},
eprint={2606.11484},
doi={10.48550/arXiv.2606.11484},
url={https://errorcorrectionzoo.org/c/simplex_15_4_8}
}
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Permanent link:
https://errorcorrectionzoo.org/c/simplex_15_4_8

Cite as:

“\([15,4,8]\) simplex code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/simplex_15_4_8, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/bits/reed_muller/dual_hamming/simplex_15_4_8.yml.