MinT - Best Known (253−103, 253, s)-Nets in Base 2

Best Known (253−103, 253, s)-Nets in Base 2

(253−103, 253, 76)-Net over F₂ — Constructive and digital

Digital (150, 253, 76)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (45, 96, 34)-net over F₂, using
  - net from sequence [i] based on digital (45, 33)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 1 place with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, 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]

(253−103, 253, 108)-Net over F₂ — Digital

Digital (150, 253, 108)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(253−103, 253, 538)-Net in Base 2 — Upper bound on s

There is no (150, 253, 539)-net in base 2, because

1 times m-reduction [i] would yield (150, 252, 539)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 7861 490243 025724 608063 686486 330543 655474 708035 685507 861974 946690 691254 738720 > 2²⁵² [i]