MinT - Best Known (223−103, 223, s)-Nets in Base 2

Best Known (223−103, 223, s)-Nets in Base 2

(223−103, 223, 59)-Net over F₂ — Constructive and digital

Digital (120, 223, 59)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (15, 66, 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, 157, 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]

(223−103, 223, 73)-Net over F₂ — Digital

Digital (120, 223, 73)-net over F₂, using

t-expansion [i] based on digital (114, 223, 73)-net over F₂, using
- net from sequence [i] based on digital (114, 72)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 114 and N(F) ≥ 73, using
    - Drinfeld modules of rank 1 [i]

(223−103, 223, 263)-Net in Base 2 — Upper bound on s

There is no (120, 223, 264)-net in base 2, because

1 times m-reduction [i] would yield (120, 222, 264)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²²², 264, S₂, 102), but
  - the linear programming bound shows that MÂ â‰¥ 973 660907 288068 814907 798296 039066 371687 485445 127151 965045 607730 242901 618708 185088 / 120 348422 834913 > 2²²² [i]