MinT - Best Known (259−98, 259, s)-Nets in Base 2

Best Known (259−98, 259, s)-Nets in Base 2

(259−98, 259, 84)-Net over F₂ — Constructive and digital

Digital (161, 259, 84)-net over F₂, using

t-expansion [i] based on digital (158, 259, 84)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (54, 104, 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, 155, 42)-net over F₂, using
    - net from sequence [i] based on digital (54, 41)-sequence over F₂ (see above)

(259−98, 259, 86)-Net in Base 2 — Constructive

(161, 259, 86)-net in base 2, using

t-expansion [i] based on (160, 259, 86)-net in base 2, using
- 1 times m-reduction [i] based on (160, 260, 86)-net in base 2, using
  - trace code for nets [i] based on (30, 130, 43)-net in base 4, using
    - net from sequence [i] based on (30, 42)-sequence in base 4, using
      - base expansion [i] based on digital (60, 42)-sequence over F₂, using
        t-expansion [i] based on digital (59, 42)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]

(259−98, 259, 131)-Net over F₂ — Digital

Digital (161, 259, 131)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(259−98, 259, 675)-Net in Base 2 — Upper bound on s

There is no (161, 259, 676)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 948541 582548 182339 431781 429562 699775 101381 161598 518237 980266 872114 383591 532904 > 2²⁵⁹ [i]