A two-distance code is a set of equal-length words whose distinct pairs have Hamming distance in a prescribed two-element set. Write for the largest size of such a code of length over a -symbol alphabet when the permitted distances are 2 and 4 (Djaljapayan et al. 2026ab). No linearity or finite field structure is assumed.

Djaljapayan et al. (2026b) proved that

for or 4 and every . For every integer , Djaljapayan et al. (2026a) proved the same formula when

where is the ceiling function. The optimal threshold is the least for which the formula holds for every . For it lies between and . Djaljapayan et al. (2026b) report ChatGPT assistance with constructing independent verification software.

For distances measured in Euclidean space, the corresponding objects are Euclidean two-distance sets. Their cardinality bounds differ from those for Hamming-space codes (Bu et al. 2026).

See also

Codeword, Error-Correcting Code, Hamming Distance

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References

Bu, B.; Lee, C.-Y.; Sheng, Y.-H.; Tsai, M.-T.; Yi, H.; and Yu, W.-H. "Johnson Optimality and Uniqueness for Distance Ratio ." 29 Oct 2026. https://arxiv.org/abs/2610.40367.Djaljapayan, V.; Lin, Y.-C. R.; and Yu, W.-H. "Asymptotically Sharp Length Thresholds for -ary Codes with Distances 2 and 4." 22 Oct 2026a. https://arxiv.org/abs/2610.30613.Djaljapayan, V.; Yu, W.-H.; and Lin, Y.-C. R. "The Ternary and Quaternary Cases of the -Distance Code Conjecture." 22 Oct 2026b. https://arxiv.org/abs/2610.30615.

Cite this as:

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

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