MinT - Best Known (225−103, 225, s)-Nets in Base 2

Best Known (225−103, 225, s)-Nets in Base 2

(225−103, 225, 59)-Net over F₂ — Constructive and digital

Digital (122, 225, 59)-net over F₂, using

3 times m-reduction [i] based on digital (122, 228, 59)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 68, 17)-net over F₂, using
    - net from sequence [i] based on digital (15, 16)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 15 and N(F) ≥ 17, using
        an explicitly constructive algebraic function field [i]
  2. digital (54, 160, 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]

(225−103, 225, 80)-Net over F₂ — Digital

Digital (122, 225, 80)-net over F₂, using

t-expansion [i] based on digital (121, 225, 80)-net over F₂, using
- net from sequence [i] based on digital (121, 79)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 121 and N(F) ≥ 80, using
    - Drinfeld modules of rank 1 [i]

(225−103, 225, 270)-Net in Base 2 — Upper bound on s

There is no (122, 225, 271)-net in base 2, because

1 times m-reduction [i] would yield (122, 224, 271)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²²⁴, 271, S₂, 102), but
  - the linear programming bound shows that MÂ â‰¥ 47 407138 423115 388183 966488 087080 744418 451432 367941 570177 125545 578341 545748 944836 886528 / 1 674805 719008 741319 > 2²²⁴ [i]