MinT - Best Known (157, 157+99, s)-Nets in Base 2

Best Known (157, 157+99, s)-Nets in Base 2

(157, 157+99, 84)-Net over F₂ — Constructive and digital

Digital (157, 256, 84)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (54, 103, 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]
2. digital (54, 153, 42)-net over F₂, using
  - net from sequence [i] based on digital (54, 41)-sequence over F₂ (see above)

(157, 157+99, 123)-Net over F₂ — Digital

Digital (157, 256, 123)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(157, 157+99, 634)-Net in Base 2 — Upper bound on s

There is no (157, 256, 635)-net in base 2, because

1 times m-reduction [i] would yield (157, 255, 635)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 58717 798706 586952 869115 969092 694527 453092 087107 717474 938668 898676 373844 258636 > 2²⁵⁵ [i]