MinT - Best Known (232−105, 232, s)-Nets in Base 2

Best Known (232−105, 232, s)-Nets in Base 2

(232−105, 232, 63)-Net over F₂ — Constructive and digital

Digital (127, 232, 63)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (21, 73, 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]
2. digital (54, 159, 42)-net over F₂, using
  - net from sequence [i] based on digital (54, 41)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
      - an explicitly constructive algebraic function field [i]

(232−105, 232, 81)-Net over F₂ — Digital

Digital (127, 232, 81)-net over F₂, using

t-expansion [i] based on digital (126, 232, 81)-net over F₂, using
- net from sequence [i] based on digital (126, 80)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 126 and N(F) ≥ 81, using
    - Drinfeld modules of rank 1 [i]

(232−105, 232, 287)-Net in Base 2 — Upper bound on s

There is no (127, 232, 288)-net in base 2, because

1 times m-reduction [i] would yield (127, 231, 288)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²³¹, 288, S₂, 104), but
  - the linear programming bound shows that MÂ â‰¥ 2321 079489 842758 634889 781394 169856 121560 276403 191987 107319 707767 179739 505587 999240 880128 / 557056 853379 564615 > 2²³¹ [i]