Here is a list of codes related to perfect codes.

[Jump to code graph excerpt]

Code Description
Combinatorial design A constant-weight binary code that is mapped into a combinatorial \(t\)-design.
Editing code A block code designed to protect against insertions, where a new symbol is added somewhere within the string, and deletions, where a symbol at an unknown location is erased.
Mixed code Encodes \(K\) states (codewords) in a string of two or more coordinates, each of which takes values in one of two or more possible groups.
Perfect binary code A type of binary code whose parameters satisfy the Hamming bound with equality.
Perfect code A type of \(q\)-ary code whose parameters satisfy the Hamming bound with equality.
Perfect quantum code A type of block quantum code whose parameters satisfy the quantum Hamming bound with equality.
\((2^{m+1}-1,2^{2n-m},3)\) Vasilyev code Member of an infinite \((2^{m+1}-1,2^{2n-m},3)\) family of perfect nonlinear codes for any \(m \geq 3\). Constructed by applying a modification of the \((u|u+v)\) construction to a perfect \((2^m-1,2^{n-m},3)\) code, not necessarily linear [1; pg. 77].
\([11,6,5]_3\) Ternary Golay code A \([11,6,5]_3\) perfect ternary linear code with connections to various areas of mathematics, e.g., lattices [2] and sporadic simple groups [1]. Adding a parity bit to the code results in the self-dual \([12,6,6]_3\) extended ternary Golay code, whose weight enumerator is the Gleason polynomial \(g_5\) [3; Rem. 4.2.6]. Up to equivalence, both codes are unique for their respective parameters [4]. The dual of the ternary Golay code is a \([11,5,6]_3\) projective two-weight subcode [5; Exam. 19.3.2].
\([23, 12, 7]\) Golay code A \([23, 12, 7]\) perfect binary linear code with connections to various areas of mathematics, e.g., lattices [2] and sporadic simple groups [1]. Up to equivalence, it is unique for its parameters [4]. The dual of the Golay code is its \([23,11,8]\) even-weight subcode [6,7].
\([2^r-1,2^r-r-1,3]\) Hamming code Member of an infinite family of perfect linear codes with parameters \([2^r-1,2^r-r-1, 3]\) for \(r \geq 2\). Their \(r \times (2^r-1) \) parity-check matrix \(H\) has all possible nonzero \(r\)-bit strings as its columns. Adding a parity check yields the \([2^r,2^r-r-1, 4]\) extended Hamming code.
\([4,2,3]_3\) Tetracode The \([4,2,3]_3\) ternary self-dual MDS code that has connections to lattices [2]. Its weight enumerator is the Gleason polynomial \(g_4\) [3; Rem. 4.2.6].
\([5,3,3]_4\) Shortened hexacode A perfect \([5,3,3]_4\) quaternary Hamming code that is the result of puncturing the hexacode [8].
\([7,4,3]\) Hamming code Second-smallest member of the Hamming code family.
\(q\)-ary Hamming code Member of an infinite family of perfect linear \(q\)-ary codes with parameters \([(q^r-1)/(q-1),(q^r-1)/(q-1)-r, 3]_q\) for \(r \geq 2\) [9; (3.1)]. These are precisely the nontrivial perfect linear codes over \(\mathbb{F}_q\) [9; Thm. 3.3.1].

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]
S. Bouyuklieva, “Self-dual codes.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[4]
P. Delsarte and J. M. Goethals, “Unrestricted codes with the golay parameters are unique”, Discrete Mathematics 12, 211 (1975) DOI
[5]
A. E. Brouwer, “Two-weight Codes.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[6]
W. Feit. Some remarks on weight functions of spaces over GF(2), unpublished (1972)
[7]
C. L. Mallows and N. J. A. Sloane, “Weight enumerators of self-orthogonal codes”, Discrete Mathematics 9, 391 (1974) DOI
[8]
G. Höhn, “Self-dual Codes over the Kleinian Four Group”, (2000) arXiv:math/0005266
[9]
P. R. J. Östergård, “Construction and Classification of Codes.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI