Here is a list of codes related to perfect codes.

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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.
Simplex integer-based code An integer-based code over the \(\ell_1\) metric whose codewords are restricted to lie in a discrete simplex. Codewords are multiplicity vectors of multisets of \(N\) symbols from a \(q\)-ary alphabet, which are invariant under a permutation channel that randomly reorders the transmitted sequence [1,2]. The \(\ell_1\) distance between codewords is the symmetric-difference distance between multisets [2].
\((2^{m+1}-1,2^{2n-m},3)\) 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 [3; 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 [4] and sporadic simple groups [3]. Adding a parity-check trit to the code yields the \([12,6,6]_3\) extended ternary Golay code. Up to equivalence, the ternary Golay code is unique for its parameters [5]. The dual of the ternary Golay code is a \([11,5,6]_3\) projective two-weight subcode [6; 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 [4] and sporadic simple groups [3]. Up to equivalence, it is unique for its parameters [5]. The dual of the Golay code is its \([23,11,8]\) even-weight subcode [7,8].
\([2^r-1,2^r-r-1,3]\) 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 [4]. It is the unique MDS code with its parameters [9,10]. Its weight enumerator is the Gleason polynomial \(g_4\) [11; 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 [12].
\([7,4,3]\) 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 \(r \geq 2\) [13; (3.1)]. These are precisely the nontrivial perfect linear codes over \(\mathbb{F}_q\) [13; Thm. 3.3.1].
List (descendants_and_cousins): All descendants and cousins of Perfect code.

References

  1. [1]

    M. Kovačević and D. Vukobratović, “Multiset Codes for Permutation Channels”, (2013) arXiv:1301.7564

  2. [2]

    M. Kovačević and D. Vukobratović, “Perfect codes in the discrete simplex”, Designs, Codes and Cryptography 75, 81 (2013) arXiv:1307.3142 DOI

  3. [3]

    F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes (Elsevier, 1977)

  4. [4]

    J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI

  5. [5]

    P. Delsarte and J. M. Goethals, “Unrestricted codes with the golay parameters are unique”, Discrete Mathematics 12, 211 (1975) DOI

  6. [6]

    A. E. Brouwer, “Two-weight Codes”, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021), pp. 449-462 DOI

  7. [7]

    W. Feit, “Some remarks on weight functions of spaces over GF(2)”, unpublished (1972)

  8. [8]

    C. L. Mallows and N. J. A. Sloane, “Weight enumerators of self-orthogonal codes”, Discrete Mathematics 9, 391 (1974) DOI

  9. [9]

    O. Taussky and J. Todd, “Covering theorems for groups”, Bulletin of the American Mathematical Society 54(3) (1948)

  10. [10]

    J. G. Kalbfleisch and R. G. Stanton, “A Combinatorial Problem in Matching”, Journal of the London Mathematical Society s1-44, 60 (1969) DOI

  11. [11]

    S. Bouyuklieva, “Self-dual codes”, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021), pp. 79-96 DOI

  12. [12]

    G. Höhn, “Self-dual Codes over the Kleinian Four Group”, (2000) arXiv:math/0005266

  13. [13]

    P. R. J. Östergård, “Construction and Classification of Codes”, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021), pp. 61-78 DOI