MinT - Best Known (228−53, 228, s)-Nets in Base 2

Best Known (228−53, 228, s)-Nets in Base 2

(228−53, 228, 200)-Net over F₂ — Constructive and digital

Digital (175, 228, 200)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 27, 5)-net over F₂, using
  - net from sequence [i] based on digital (1, 4)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 1 and N(F) ≥ 5, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (148, 201, 195)-net over F₂, using
  - trace code for nets [i] based on digital (14, 67, 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]

(228−53, 228, 379)-Net over F₂ — Digital

Digital (175, 228, 379)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(228−53, 228, 4444)-Net in Base 2 — Upper bound on s

There is no (175, 228, 4445)-net in base 2, because

1 times m-reduction [i] would yield (175, 227, 4445)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 216 794619 327941 069562 023065 757411 143227 052189 513443 292704 397104 333694 > 2²²⁷ [i]