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

Alternative names: RM\(^*(1,3)\) code, Little Hamming code.

Description

Second-smallest nontrivial member of the simplex-code family. The columns of its generator matrix are in one-to-one correspondence with the elements of the projective space \(PG(2,2)\), with each column being a chosen representative of the corresponding element. The codewords form a \((8,9)\) simplex spherical code under the antipodal mapping. As a simplex code, it is equidistant: every nonzero codeword has Hamming weight \(4\).

Its generator matrix is \begin{align} \left(\begin{array}{ccccccc} 1 & 0 & 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 & 0 & 1 \\ \end{array}\right)~. \tag*{(1)}\end{align} The automorphism group of the code is \(GL(3,\mathbb{F}_2)\cong PSL(2,\mathbb{F}_7)\), the second-smallest non-abelian finite simple group.

Protection

Being a simplex code, it saturates the Plotkin bound [1; pg. 43].

Cousins

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
\([2^m-1,m,2^{m-1}]\) simplex codeGRM Evaluation MDS Projective geometry Linear \(q\)-ary LCC LRC Distributed-storage LDC AG OA Universally optimal ECC \(t\)-design
Parents
\([2^m-1,m,2^{m-1}]\) simplex code
Difference-set cyclic (DSC) code\(q\)-ary LDPC Tanner Linear \(q\)-ary Linear code over \(\mathbb{Z}_q\) LRC Distributed-storage Cyclic Skew-cyclic Constacyclic ECC
The \([7,3,4]\) simplex code is the smallest difference-set cyclic code, arising from the lines of the projective plane \(PG(2,2)\) [1; pg. 397].
Incidence-matrix projective codeProjective geometry Linear \(q\)-ary ECC
The \([7,3,4]\) simplex code is the smallest difference-set cyclic code, arising from the lines of the projective plane \(PG(2,2)\) [1; pg. 397].
Small-distance block codeECC
\([7,3,4]\) simplex code

References

[1]
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes (Elsevier, 1977)
[2]
W. Feit. Some remarks on weight functions of spaces over GF(2), unpublished (1972)
[3]
C. L. Mallows and N. J. A. Sloane, “Weight enumerators of self-orthogonal codes”, Discrete Mathematics 9, 391 (1974) DOI
[4]
T. Ericson and V. Zinoviev, eds., Codes on Euclidean Spheres (Elsevier, 2001)
[5]
J. H. Conway and N. J. A. Sloane, “On the Voronoi Regions of Certain Lattices”, SIAM Journal on Algebraic Discrete Methods 5, 294 (1984) DOI
[6]
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
[7]
M. Shi, T. Honold, P. Sole, Y. Qiu, R. Wu, and Z. Sepasdar, “The Geometry of Two-Weight Codes Over ℤ\({}_{\text{\textit{p}}}\)\({}^{\text{\textit{m}}}\)”, IEEE Transactions on Information Theory 67, 7769 (2021) DOI
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Zoo Code ID: simplex734

Cite as:
“\([7,3,4]\) simplex code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/simplex734
BibTeX:
@incollection{eczoo_simplex734, title={\([7,3,4]\) simplex code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/simplex734} }
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Cite as:

“\([7,3,4]\) simplex code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/simplex734

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/bits/reed_muller/dual_hamming/simplex734.yml.