MinT - Best Known (6, 17, s)-Nets in Base 3

Best Known (6, 17, s)-Nets in Base 3

(6, 17, 14)-Net over F₃ — Constructive and digital

Digital (6, 17, 14)-net over F₃, using

net from sequence [i] based on digital (6, 13)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 6 and N(F) ≥ 14, using
  - an explicitly constructive algebraic function field [i]

(6, 17, 32)-Net over F₃ — Upper bound on s (digital)

There is no digital (6, 17, 33)-net over F₃, because

2 times m-reduction [i] would yield digital (6, 15, 33)-net over F₃, but
- extracting embedded orthogonal array [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]

(6, 17, 39)-Net in Base 3 — Upper bound on s

There is no (6, 17, 40)-net in base 3, because

1 times m-reduction [i] would yield (6, 16, 40)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 45 576657 > 3¹⁶ [i]