For every integer n≥ 6 and every integer k with 3≤ k≤ n-3, there exists a k-dimensional F_2-subspace E⊆F_2^n such that displaystyle∑_x∈ E setminus0x^-1=0.

Carlet’s conjecture on zero-sum subspaces for the multiplicative inverse

Carlet’s conjecture asks whether, for every $3\le k\le n-3$ , there is a $k$ -dimensional subspace $E$ of $\mathbb{F}_{2^n}$ whose nonzero inverse elements sum to zero.

Known results

Hou and Zhao (2025) proved the conjecture when $n$ is not prime.
Absolute-irreducibility results establish further ranges and confirm all odd $n\le 27$ , with exceptions only in specified cases when $7\mid n$ .
For odd prime $n$ , the known range reaches $3\le k\le \frac{3}{13}n+0.461$ .
For even $n$ , the inverse function is known not to be $k$ th-order sum-free throughout $2\le k\le n-2$ .

October 2026 claimed proof

Kaimin Cheng’s preprint combines asymptotic counting, low-dimensional cases, duality, and subfield constructions to claim existence in every admissible dimension. It is unrefereed and has no independent mathematical corroboration in the retrieved sources.

Current status (as of October 2026): Cheng claims a complete proof, but it remains unverified; absent independent confirmation, the conjecture is not established.

Carlet’s conjecture on zero-sum subspaces for the multiplicative inverse

Primary source

Additional references

Progress summary

Solutions 0