MinT - Best Known (256−59, 256, s)-Nets in Base 2

Best Known (256−59, 256, s)-Nets in Base 2

(256−59, 256, 206)-Net over F₂ — Constructive and digital

Digital (197, 256, 206)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (8, 37, 11)-net over F₂, using
  - net from sequence [i] based on digital (8, 10)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 8 and N(F) ≥ 11, using
      - an explicitly constructive algebraic function field [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]

(256−59, 256, 429)-Net over F₂ — Digital

Digital (197, 256, 429)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(256−59, 256, 5134)-Net in Base 2 — Upper bound on s

There is no (197, 256, 5135)-net in base 2, because

1 times m-reduction [i] would yield (197, 255, 5135)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 58044 971205 497784 284517 688564 039284 373869 577113 398671 789050 754417 064021 398272 > 2²⁵⁵ [i]