MinT - Best Known (38, 38+37, s)-Nets in Base 2

Best Known (38, 38+37, s)-Nets in Base 2

(38, 38+37, 26)-Net over F₂ — Constructive and digital

Digital (38, 75, 26)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (9, 27, 12)-net over F₂, using
  - net from sequence [i] based on digital (9, 11)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 9 and N(F) ≥ 12, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (11, 48, 14)-net over F₂, using
  - net from sequence [i] based on digital (11, 13)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 11 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]

(38, 38+37, 30)-Net over F₂ — Digital

Digital (38, 75, 30)-net over F₂, using

t-expansion [i] based on digital (36, 75, 30)-net over F₂, using
- net from sequence [i] based on digital (36, 29)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 36 and N(F) ≥ 30, using
    - Drinfeld modules of rank 1 [i]

(38, 38+37, 91)-Net over F₂ — Upper bound on s (digital)

There is no digital (38, 75, 92)-net over F₂, because

1 times m-reduction [i] would yield digital (38, 74, 92)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁷⁴, 92, F₂, 36) (dual of [92, 18, 37]-code), but
  - adding a parity check bit [i] would yield linear OA(2⁷⁵, 93, F₂, 37) (dual of [93, 18, 38]-code), but
    - â€œBroâ€ bound on codes from Brouwerâ€™s database [i]

(38, 38+37, 93)-Net in Base 2 — Upper bound on s

There is no (38, 75, 94)-net in base 2, because

1 times m-reduction [i] would yield (38, 74, 94)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁷⁴, 94, S₂, 36), but
  - the linear programming bound shows that MÂ â‰¥ 2115 166837 143245 569795 293184 / 83421 > 2⁷⁴ [i]