MinT - Best Known (58−27, 58, s)-Nets in Base 2

Best Known (58−27, 58, s)-Nets in Base 2

(58−27, 58, 25)-Net over F₂ — Constructive and digital

Digital (31, 58, 25)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (7, 20, 11)-net over F₂, using
  - linear code embedding found in a computer search [i]
2. digital (11, 38, 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]

(58−27, 58, 27)-Net over F₂ — Digital

Digital (31, 58, 27)-net over F₂, using

net from sequence [i] based on digital (31, 26)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 31 and N(F) ≥ 27, using
  - Drinfeld modules of rank 1 [i]

(58−27, 58, 96)-Net over F₂ — Upper bound on s (digital)

There is no digital (31, 58, 97)-net over F₂, because

1 times m-reduction [i] would yield digital (31, 57, 97)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2⁵⁷, 97, F₂, 26) (dual of [97, 40, 27]-code), but
  - adding a parity check bit [i] would yield linear OA(2⁵⁸, 98, F₂, 27) (dual of [98, 40, 28]-code), but
    - â€œBroâ€ bound on codes from Brouwerâ€™s database [i]

(58−27, 58, 97)-Net in Base 2 — Upper bound on s

There is no (31, 58, 98)-net in base 2, because

1 times m-reduction [i] would yield (31, 57, 98)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁵⁷, 98, S₂, 26), but
  - the linear programming bound shows that MÂ â‰¥ 796 660189 281003 951121 817121 767504 740352 / 5287 278143 181290 578125 > 2⁵⁷ [i]