\([16,5,8]\) first-order RM code

\([16,5,8]\) first-order RM code

Description

Binary code that corrects three bit errors on sixteen bits and from which the NR code, the \(\Lambda_{16}\) Barnes-Wall lattice, and the tesseract color code are constructed. It is the \(m=4\) member of the first-order RM family.

Cousins

Binary BCH code— Extending the \([15,5,7]\) BCH code by a parity check yields the \([16,5,8]\) first-order RM code [1; Ch. 13, Thm. 11].
\(\Lambda_{16}\) Barnes-Wall lattice— Applying Construction B to the \([16,5,8]\) first-order RM code yields the \(\Lambda_{16}\) Barnes-Wall lattice [2; Ch. 4, pgs. 129-130][3; Exam. 10.7.2].
\([24, 12, 8]\) Extended Golay code— The \([16,5,8]\) first-order RM code is a subcode of the extended Golay code [2; Ch. 5, pg. 146].
\((16,256,6)\) Nordstrom-Robinson (NR) code— The NR code is the union of eight cosets of the \([16,5,8]\) first-order RM code [1; pgs. 74 and 476].
\([[16,6,4]]\) Tesseract color code— The tesseract color code is constructed from the \([16,5,8]\) first-order RM code via the CSS construction [4,5].

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

Coxeter code

Reed-Muller (RM) codeLinear code over \(\mathbb{Z}_q\) GRM Evaluation Divisible Linear \(q\)-ary AG LCC LRC Distributed-storage ECC

\([2^m,m+1,2^{m-1}]\) First-order RM codeProjective geometry Linear \(q\)-ary ECC

Parents

\([2^m,m+1,2^{m-1}]\) First-order RM code

The \([16,5,8]\) first-order RM code is the RM\((1,4)\) code.

\([16,5,8]\) first-order RM code

References

[1]: F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes (Elsevier, 1977)
[2]: J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
[3]: T. Ericson and V. A. Zinoviev, eds., Codes on Euclidean Spheres (Elsevier, 2001)
[4]: J. Haah, M. B. Hastings, D. Poulin, and D. Wecker, “Magic state distillation with low space overhead and optimal asymptotic input count”, Quantum 1, 31 (2017) arXiv:1703.07847 DOI
[5]: A. Gong and J. M. Renes, “Computation with quantum Reed-Muller codes and their mapping onto 2D atom arrays”, (2024) arXiv:2410.23263

Page edit log

Victor V. Albert (2026-10-13) — most recent

Cite as:

“\([16,5,8]\) first-order RM code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/rm_16_5_8, arXiv:2606.11484

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/bits/reed_muller/rm_16_5_8.yml.

\([16,5,8]\) first-order RM code

Description

Cousins

Member of code lists

Primary Hierarchy

References

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