MinT - Best Known (16, 16+30, s)-Nets in Base 3

Best Known (16, 16+30, s)-Nets in Base 3

(16, 16+30, 28)-Net over F₃ — Constructive and digital

Digital (16, 46, 28)-net over F₃, using

t-expansion [i] based on digital (15, 46, 28)-net over F₃, using
- net from sequence [i] based on digital (15, 27)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
    - an explicitly constructive algebraic function field [i]

(16, 16+30, 64)-Net over F₃ — Upper bound on s (digital)

There is no digital (16, 46, 65)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3⁴⁶, 65, F₃, 30) (dual of [65, 19, 31]-code), but
- residual code [i] would yield linear OA(3¹⁶, 34, F₃, 10) (dual of [34, 18, 11]-code), but
  - 1 times truncation [i] would yield linear OA(3¹⁵, 33, F₃, 9) (dual of [33, 18, 10]-code), but
    - â€œLPâ€ bound on codes from Brouwerâ€™s database [i]

(16, 16+30, 67)-Net in Base 3 — Upper bound on s

There is no (16, 46, 68)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3⁴⁶, 68, S₃, 30), but
- the linear programming bound shows that MÂ â‰¥ 755205 235334 095085 530724 844560 924379 / 84 114825 914875 > 3⁴⁶ [i]