MathDB

The nested-clique B-form code distance conjecture

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Let tt be an odd prime number. Consider the nested clique graph with respect to the permutations in beqrefbpermsd, and let bmathcalCdbmathcal Cd be the associated BB-form code. Its length is the number of coordinates in the construction. Nested-clique B-form code conjecture. The code bmathcalCdbmathcal Cd has length t2t^2 and binary distance at least 2t−22t-2. The conjecture would imply a family of codes whose binary distance grows on the order of the square root of the length, since the construction has length t2t^2; the source states that this remains to be proved or disproved.

References

Primary source

Yuma Furuta, “Relation between spectra of Narain CFTs and properties of associated boolean functions”, arXiv:2203.11643 (2022).

coding-theorygraph-theoryquantum-error-correcting-codes
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Progress summary

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Claimed progress

The conjecture is supported for some prime sizes, but no public proof or counterexample is known for all odd primes.

The conjecture asserts that the nested-clique BB-form code always has length t2t^2 and binary distance at least 2t−22t-2 for every odd prime tt. Its stated coding-theoretic consequence is a family with distance growing like the square root of its length.

Partial constructions (date not stated)

Yuma Furuta's paper constructs nested-clique examples with length t2t^2 and binary distance 2t−22t-2 for some prime values of tt, and identifies this distance with the associated Boolean function's EPC distance. This supports the conjectured parameters but does not establish them for every odd prime.

Current status (as of August 2026): The conjecture is unproved in general, with no reported counterexample; the available evidence verifies the claimed parameters only for some prime values of tt.

Sources

Solutions 0

No solutions have been posted yet.