MinT - Best Known (9, 40, s)-Nets in Base 4

Best Known (9, 40, s)-Nets in Base 4

(9, 40, 22)-Net over F₄ — Constructive and digital

Digital (9, 40, 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, 40, 26)-Net over F₄ — Digital

Digital (9, 40, 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, 40, 44)-Net over F₄ — Upper bound on s (digital)

There is no digital (9, 40, 45)-net over F₄, because

1 times m-reduction [i] would yield digital (9, 39, 45)-net over F₄, but
- 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, 40, 45)-Net in Base 4 — Upper bound on s

There is no (9, 40, 46)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁴⁰, 46, S₄, 31), but
- the linear programming bound shows that MÂ â‰¥ 976 812062 248620 373162 590208 / 703 > 4⁴⁰ [i]