MinT - Best Known (253−59, 253, s)-Nets in Base 2

Best Known (253−59, 253, s)-Nets in Base 2

(253−59, 253, 204)-Net over F₂ — Constructive and digital

Digital (194, 253, 204)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (5, 34, 9)-net over F₂, using
  - net from sequence [i] based on digital (5, 8)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 5 and N(F) ≥ 9, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (160, 219, 195)-net over F₂, using
  - trace code for nets [i] based on digital (14, 73, 65)-net over F₈, using
    - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
        the Suzuki function field over F₈ [i]

(253−59, 253, 411)-Net over F₂ — Digital

Digital (194, 253, 411)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(253−59, 253, 4776)-Net in Base 2 — Upper bound on s

There is no (194, 253, 4777)-net in base 2, because

1 times m-reduction [i] would yield (194, 252, 4777)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 7267 223883 156406 302688 080199 050905 818404 151787 276743 880553 478653 821315 256304 > 2²⁵² [i]