Binary code 

Root code for the Binary Kingdom

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 \(\mathbb{Z}_2\wr S_n=\mathbb{Z}_2^n\rtimes S_n\), 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 the codewords of one code can be mapped into those of the other under a hyperoctahedral group element [4][3; Def. 1.8.8]. Determining equivalence of two codes can be done by putting each in a canonical form and mapping to a graph isomorphism problem [5].

Protection

A binary code \(C\) corrects \(t\) errors in the Hamming distance if \begin{align} \forall x \in C~,~D(x,x+y) < D(x' , x+y) \tag*{(1)}\end{align} for all codewords \(x' \neq x\) and all \(y\) such that \(|y|=t\), where \(D\) is the Hamming distance and \(|y| = D(y,0) \). A code corrects \(t\) errors if and only if \(d \geq 2t+1\), i.e., a code corrects errors on \(t \leq \left\lfloor (d-1)/2 \right\rfloor\) coordinates. The number of correctable errors is called the decoding radius, and it is upper bounded by half of the distance. In addition, a code detects errors on up to \(d-1\) coordinates, and corrects erasure errors on up to \(d-1\) coordinates.

Rate

The rate of a binary code is usually defined as \(R=\frac{1}{n}\log_{2}K\) bits per symbol. The maximum rate of a binary code with linear distance is upper bounded by the McEliece, Rodemich, Rumsey and Welch (MRRW) bound [6] (see Refs. [7–10] for other proofs).

Gates

The group of reversible binary operations is the permutation group \(S_{2^n}\) of all possible \(n\)-bit strings. Reversible gate sets have been classified [11].

Decoding

For few-bit codes (\(n\) is small), decoding can be based on a lookup table. For infinite code families, the size of such a table scales exponentially with \(n\), so approximate decoding algorithms scaling polynomially with \(n\) have to be used. The decoder determining the most likely error given a noise channel is called the maximum-likelihood decoder.Given a received string \(x\) and an error bound \(e\), a list decoder returns a list of all codewords that are at most \(e\) from \(x\) in Hamming distance. The number of codewords in a neighborhood of \(x\) has to be polynomial in \(n\) in order for this decoder to run in time polynomial in \(n\).

Parent

Children

Cousins

  • Orthogonal array (OA) — An \((n,K)\) binary code with dual distance \(d^{\perp}\) is an OA\(_{K/2^{d^{\perp}-1}}(d^{\perp}-1,n,2)\) [12][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.
  • Sum-rank-metric code — The sum-rank metric generalizes both the Hamming metric and the rank metric [13].
  • Hypercube code — Binary strings are elements of the Hamming \(n\)-cube (a.k.a. Boolean hypercube).
  • 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.
  • Symmetry-protected topological (SPT) code — SPT orders may be used for encoding classical information [14].
  • 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. Any self-complementary \((n,K,d > 1)\) classical code yields an \(((n,K/2,2))\) self-complementary quantum code whose quantum codewords are superpositions of the classical codewords and their complements [15; Lemma 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]
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]
Classification Algorithms for Codes and Designs (Springer-Verlag, 2006) DOI
[6]
R. McEliece, E. Rodemich, H. Rumsey, and L. Welch, “New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities”, IEEE Transactions on Information Theory 23, 157 (1977) DOI
[7]
M. Navon and A. Samorodnitsky, “Linear Programming Bounds for Codes via a Covering Argument”, Discrete &amp; Computational Geometry 41, 199 (2008) DOI
[8]
J. Friedman and J.-P. Tillich, “Generalized Alon--Boppana Theorems and Error-Correcting Codes”, SIAM Journal on Discrete Mathematics 19, 700 (2005) DOI
[9]
A. Samorodnitsky, “One more proof of the first linear programming bound for binary codes and two conjectures”, (2021) arXiv:2104.14587
[10]
N. Linial and E. Loyfer, “An Elementary Proof of the First LP Bound on the Rate of Binary Codes”, (2023) arXiv:2303.16619
[11]
S. Aaronson, D. Grier, and L. Schaeffer, “The Classification of Reversible Bit Operations”, (2015) arXiv:1504.05155
[12]
Delsarte, Philippe. "Bounds for unrestricted codes, by linear programming." (1972).
[13]
U. Martínez-Peñas, “Skew and linearized Reed-Solomon codes and maximum sum rank distance codes over any division ring”, (2018) arXiv:1710.03109
[14]
B. Zeng and D.-L. Zhou, “Topological and Error-Correcting Properties for Symmetry-Protected Topological Order”, (2014) arXiv:1407.3413
[15]
J. A. Smolin, G. Smith, and S. Wehner, “Simple Family of Nonadditive Quantum Codes”, Physical Review Letters 99, (2007) arXiv:quant-ph/0701065 DOI
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Zoo Code ID: bits_into_bits

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“Binary code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/bits_into_bits
BibTeX:
@incollection{eczoo_bits_into_bits, title={Binary code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/bits_into_bits} }
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“Binary code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/bits_into_bits

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