Root families for optimal ternary cyclic codes: Ding–Helleseth's Open Problem 7.11 for m ≤ 16, and why the exception at (m,h)=(10,8) is not sporadic

Abstract

For q=3ᵐ and n=q-1, let C_((1,e)) be the ternary cyclic code of length n with generator polynomial m_α(x)m_(αᵉ)(x). Ding and Helleseth reduced the question of when C_((1,e)) attains the sphere-packing-optimal parameters [n, n-2m, 4] to two root conditions on the polynomials (x+1)ᵉ ± xᵉ ∓ 1 over 𝔽_q, and left nine open problems; the fifth of these asks for the conditions on m and h under which e=(3ʰ-5)/2 is optimal for even m. We answer it for every even m ≤ 16, and then explain the answer. The computed rows have exactly one irregular cell, (m,h)=(10,8). We show that this cell is not sporadic: for every h ≡ 8 (mod 10) the equation (x+1)ᵉ-xᵉ-1=0 has the same eleven solutions in 𝔽₃¹⁰, ten of them of degree exactly 10 over 𝔽₃, so the criterion fails whenever 10 | m. The next cell this reaches, (m,h)=(20,18), lies beyond every computed row and has h ≡ 2 (mod 4), which refutes the congruence pattern that the computed rows alone suggest. We record the complementary family — for h ≡ 0 (mod 4) an extra root already lives in the 81-element field, hence the criterion fails in 𝔽_(3ᵐ) for every m ≡ 0 (mod 4) — and a sweep of the subfield lattice showing that any third family needs a root of degree at least 13. Along the way we answer Bao and Zou's Problem 6.2 at m=2,6,10,14, where exactly 2, 6, 20 and 42 of the admissible exponents are optimal, and tabulate Ding–Helleseth's Open Problem 7.15 at every odd m ≤ 13, where the top exponent h=m-1 always fails. All statements are machine-checked in Lean 4, apart from five steps — two elementary arguments, two external computations, and the identification of the fast search engine with the criterion — each flagged where it occurs.

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