MinT - Best Known (9, 39, s)-Nets in Base 4

Best Known (9, 39, s)-Nets in Base 4

(9, 39, 22)-Net over F₄ — Constructive and digital

Digital (9, 39, 22)-net over F₄, using

net from sequence [i] based on digital (9, 21)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 9 and N(F) ≥ 22, using
  - an explicitly constructive algebraic function field [i]

(9, 39, 26)-Net over F₄ — Digital

Digital (9, 39, 26)-net over F₄, using

net from sequence [i] based on digital (9, 25)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 9 and N(F) ≥ 26, using
  - Drinfeld modules of rank 1 [i]

(9, 39, 44)-Net over F₄ — Upper bound on s (digital)

There is no digital (9, 39, 45)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4³⁹, 45, F₄, 30) (dual of [45, 6, 31]-code), but
- â€œDaâ€ bound on codes from Brouwerâ€™s database [i]

(9, 39, 47)-Net in Base 4 — Upper bound on s

There is no (9, 39, 48)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4³⁹, 48, S₄, 30), but
- the linear programming bound shows that MÂ â‰¥ 158456 325028 528675 187087 900672 / 329189 > 4³⁹ [i]