Description
Encodes \(K\) states (codewords) in \(n\) binary coordinates and has distance \(d\). Usually denoted as \((n,K,d)\). The distance is the minimum Hamming distance between a pair of distinct codewords.
The coordinate permutation group \(S_n\) of order \(n!\) is formed by \(n\)-dimensional matrices with a 1 in each row and column [1; Ch. 8][2; Ch. 3]. The group of isometries of Hamming space is the hyperoctahedral group \(\Z_2\wr S_n=\Z_2^n\rtimes S_n\), i.e., the permutation group together with the group formed by the action of binary space on itself (under addition). Two binary codes are equivalent if there exists a hyperoctahedral group element that maps one into the other [4][3; Def. 1.8.8].
Protection
Rate
Decoding
Parent
Children
- Binary group-orbit code
- Combinatorial design — If the number of a code is less than or equal to its dual distance, then some sets of fixed-weight codewords form a combinatorial design [1; Thm. 6.7].
- Nearly perfect code
- Unary code
- Anticode
- Batch code
- Conference code
- Constantin-Rao (CR) code
- Hergert code
- Gray code
- Delsarte-Goethals (DG) code
- Levenshtein code
- Best \((10,40,4)\) code
- Sloane-Whitehead code
- Vasilyev code
Cousins
- Orthogonal array (OA) — An \((n,K)\) binary code with dual distance \(d^{\perp}\) is an OA\(_{K/2^{d^{\perp}-1}}(d^{\perp}-1,n,2)\) [10][1; Ch. 5].
- Qubit c-q code — Any binary code can be embedded into a qubit Hilbert space, and thus passed through a qubit channel, by associating length-\(n\) bitstrings with basis vectors in a Hilbert space over \(\mathbb{Z}_2^n\). For example, a bit of information can be embedded into a two-dimensional vector space by associating the two bit values with two basis vectors for the space.
- Construction-\(A\) code — Each binary code yields a sphere packing under Construction A.
- Binary antipodal code — Binary antipodal codes are spherical codes obtained from binary codes via the antipodal mapping.
- Fock-state bosonic code — Fock-state code distance is a natural extension of Hamming distance between binary strings.
- Movassagh-Ouyang Hamiltonian code — Movassagh-Ouyang codes are constructed from classical binary codes.
- Self-complementary quantum code — A binary code is called self-complementary if, for each codeword \(c\), its negation \(\overline{c}\) is also a codeword.
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]
- W. C. Huffman, J.-L. Kim, and P. Solé, "Basics of coding theory." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [4]
- P. R. J. Östergård, "Construction and Classification of Codes." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [5]
- R. McEliece et al., “New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities”, IEEE Transactions on Information Theory 23, 157 (1977) DOI
- [6]
- M. Navon and A. Samorodnitsky, “Linear Programming Bounds for Codes via a Covering Argument”, Discrete & Computational Geometry 41, 199 (2008) DOI
- [7]
- J. Friedman and J.-P. Tillich, “Generalized Alon--Boppana Theorems and Error-Correcting Codes”, SIAM Journal on Discrete Mathematics 19, 700 (2005) DOI
- [8]
- A. Samorodnitsky, “One more proof of the first linear programming bound for binary codes and two conjectures”, (2021) arXiv:2104.14587
- [9]
- N. Linial and E. Loyfer, “An Elementary Proof of the First LP Bound on the Rate of Binary Codes”, (2023) arXiv:2303.16619
- [10]
- Delsarte, Philippe. "Bounds for unrestricted codes, by linear programming." (1972).
Page edit log
- Victor V. Albert (2022-07-06) — most recent
- Victor V. Albert (2022-02-16)
- Victor V. Albert (2021-10-29)
Cite as:
“Binary code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/bits_into_bits
Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/bits/bits_into_bits.yml.