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Binary Code

A binary code of length n is a set C subset= {0,1}^n whose elements are called codewords. The Hamming distance d_H(x,y) between two codewords is the number of coordinates in which they differ. Let

A_2(n,d)=max{|C|:C subset= {0,1}^n, d_H(x,y)>=d for distinct x,y in C}.

Thus, A_2(n,d) is the largest size of a binary code of length n having minimum distance at least d. For fixed 0<delta<1/2, its asymptotic rate is

R_2(delta)=limsup_(n->infty)1/nlog_2A_2(n,[deltan]).

Let M_2(delta) denote the optimized McEliece-Rodemich-Rumsey-Welch bound exponent and kappa_(bin)(delta) the new upper-bound exponent constructed by OpenAI (2026). The AI-generated proof establishes R_2(delta)<=kappa_(bin)(delta)<M_2(delta) for every fixed 0<delta<1/2. This is the first improvement to the general high-dimensional binary-code exponent since 1977.

See also

Binary, Closest Vector Problem, Coding Theory, Error-Correcting Code, Hamming Code, Hamming Distance, Linear Code, McEliece-Rodemich-Rumsey-Welch Bound

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References

MacWilliams, F. J. and Sloane, N. J. A. The Theory of Error-Correcting Codes. Amsterdam, Netherlands: North-Holland, 1977.McEliece, R. J.; Rodemich, E. R.; Rumsey, H. C. Jr.; and Welch, L. R. "New Upper Bounds on the Rate of a Code via the Delsarte-MacWilliams Inequalities." IEEE Trans. Inform. Th. 23, 157-166, 1977. https://doi.org/10.1109/TIT.1977.1055688.OpenAI. "Improved Bounds for Binary and Spherical Codes." Ch. 2 in Ten Advances in Mathematics and Theoretical Computer Science. Aug. 1, 2026. https://cdn.openai.com/pdf/ten-proofs-oai.pdf.

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Binary Code

Cite this as:

Weisstein, Eric W. "Binary Code." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BinaryCode.html

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