AN EXACT ROOT COUNT IN A PROBLEM ON OPTIMAL TERNARY CYCLIC CODES

Abstract Let q = 3 m $q=3^m$ q equals 3 Superscript m with m $m$ m odd and let s ≥ 1 $s\\geq 1$ s greater than or equals 1 . We determine exactly the number of roots in F q $\\mathbb {F}_q$ double struck upper F Subscript q of x 3 s + 1 − x 2 + 1 = 0. $ x^{3^s+1}-x^2+1=0. $ x Superscript 3 Super Superscript s Superscript plus 1 Baseline minus x squared plus 1 equals 0 period This equation appears in the study of open problems of Ding and Helleseth [‘Optimal ternary cyclic codes from monomials’, IEEE Trans. Inform. Theory 59 (9) (2013), 5898–5904]. A Cayley transform sends the equation to the norm-one subgroup of F q 2 × $\\mathbb {F}_{q^2}^{\\times }$ double struck upper F Subscript q squared Superscript times , where the question becomes a calculation in a cyclic group. We prove that # { x ∈ F q : x