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Ward's Conjecture

Ward's conjecture is a divisibility assertion about Griesmer codes. For an [n,k,d]_q Griesmer code, write q=p^f with p a prime number, and let e=nu_p(d), where nu_p(d) is the greatest dividing exponent of p in d. The conjecture asserts that the Hamming weight of every codeword is divisible by p^(max(0,e-f+1)) (Deng and Zhang 2026).

For a prime field, this gives divisibility by p^e. Over a proper extension field, the exponent need not be e. Deng and Zhang (2026) proved the conjecture.

See also

Griesmer Code, Hamming Weight

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References

Deng, H. and Zhang, W. "A Proof of Ward's Conjecture on Griesmer Codes." 21 Oct 2026. https://arxiv.org/abs/2610.29275.

Cite this as:

Weisstein, Eric W. "Ward's Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WardsConjecture.html

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