Let q=5^m with m≥3 odd. Let C_(q,q+1,4,1) be the BCH code appearing in the source, and let overline C_4 be the extended classical MDS cyclic code. The minimum-distance conjecture. Then d( C_(q,q+1,4,1))=5 and d((overline C_4)^perp)=6. This is the remaining open case identified by the source for the minimum distance of (overline C_4)^perp; it concerns the distance of a BCH code and its dual extended-code counterpart.

The minimum-distance conjecture for extended codes over fields of characteristic five

Let $q=5^m$ with $m\geq3$ odd. Let $\mathcal C_{(q,q+1,4,1)}$ be the BCH code appearing in the source, and let $\overline{\mathcal C}_4$ be the extended classical MDS cyclic code. The minimum-distance conjecture. Then

d(\mathcal C_{(q,q+1,4,1)})=5

and

d((\overline{\mathcal C}_4)^\perp)=6.

This is the remaining open case identified by the source for the minimum distance of $(\overline{\mathcal C}_4)^\perp$ ; it concerns the distance of a BCH code and its dual extended-code counterpart.

Sources & referencesView supporting material

Primary source

Zhonghua Sun and Cunsheng Ding, “The extended codes of a family of reversible MDS cyclic codes”, arXiv:2307.04076 (2023).

Additional references

8 papers in this index state this conjecture (2011–2023). The statement above is taken from the most recent of them; the others are arXiv:2202.06237, arXiv:2008.09935, arXiv:1911.11738, arXiv:1608.02169, arXiv:1608.02170, arXiv:1608.02670, arXiv:1112.0283.

1 week ago

Progress summary

Refreshed 5 days ago

Open

The conjecture remains unproved: it predicts two exact minimum distances for related error-correcting codes over certain fields of characteristic five.

For field sizes that are odd powers of five with exponent at least three, the conjecture asserts the two distances stated in the problem. The source identifies this as the remaining unresolved case and records their relation as $d((\overline{\mathcal C}_4)^\perp)=d(\mathcal C_{(q,q+1,4,1)})+1$ .

Current status (as of August 2026): The conjecture is open; no proof, counterexample, verification, or claimed solution was found in the retrieved sources.

No solutions have been posted yet.