MinT - Best Known (47−23, 47, s)-Nets in Base 2

Best Known (47−23, 47, s)-Nets in Base 2

(47−23, 47, 21)-Net over F₂ — Constructive and digital

Digital (24, 47, 21)-net over F₂, using

t-expansion [i] based on digital (21, 47, 21)-net over F₂, using
- net from sequence [i] based on digital (21, 20)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 21 and N(F) ≥ 21, using
    - an explicitly constructive algebraic function field [i]

(47−23, 47, 22)-Net over F₂ — Digital

Digital (24, 47, 22)-net over F₂, using

t-expansion [i] based on digital (23, 47, 22)-net over F₂, using
- net from sequence [i] based on digital (23, 21)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 23 and N(F) ≥ 22, using
    - Drinfeld modules of rank 1 [i]

(47−23, 47, 63)-Net over F₂ — Upper bound on s (digital)

There is no digital (24, 47, 64)-net over F₂, because

1 times m-reduction [i] would yield digital (24, 46, 64)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁴⁶, 64, F₂, 22) (dual of [64, 18, 23]-code), but
  - â€œJaâ€ bound on codes from Brouwerâ€™s database [i]

(47−23, 47, 71)-Net in Base 2 — Upper bound on s

There is no (24, 47, 72)-net in base 2, because

1 times m-reduction [i] would yield (24, 46, 72)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁴⁶, 72, S₂, 22), but
  - the linear programming bound shows that MÂ â‰¥ 819 958843 681301 069824 / 9 976365 > 2⁴⁶ [i]