MinT - Best Known (253−113, 253, s)-Nets in Base 2

Best Known (253−113, 253, s)-Nets in Base 2

(253−113, 253, 67)-Net over F₂ — Constructive and digital

Digital (140, 253, 67)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (39, 95, 33)-net over F₂, using
  - net from sequence [i] based on digital (39, 32)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 39 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (45, 158, 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]

(253−113, 253, 88)-Net over F₂ — Digital

Digital (140, 253, 88)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(253−113, 253, 413)-Net in Base 2 — Upper bound on s

There is no (140, 253, 414)-net in base 2, because

1 times m-reduction [i] would yield (140, 252, 414)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 7520 295480 617470 983144 036988 852245 741980 112528 695082 897723 279125 250099 358900 > 2²⁵² [i]