MinT - Best Known (232−87, 232, s)-Nets in Base 2

Best Known (232−87, 232, s)-Nets in Base 2

(232−87, 232, 77)-Net over F₂ — Constructive and digital

Digital (145, 232, 77)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (48, 91, 35)-net over F₂, using
  - net from sequence [i] based on digital (48, 34)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 2 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]
2. digital (54, 141, 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]

(232−87, 232, 84)-Net in Base 2 — Constructive

(145, 232, 84)-net in base 2, using

4 times m-reduction [i] based on (145, 236, 84)-net in base 2, using
- trace code for nets [i] based on (27, 118, 42)-net in base 4, using
  - net from sequence [i] based on (27, 41)-sequence in base 4, using
    - base expansion [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]

(232−87, 232, 121)-Net over F₂ — Digital

Digital (145, 232, 121)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(232−87, 232, 637)-Net in Base 2 — Upper bound on s

There is no (145, 232, 638)-net in base 2, because

1 times m-reduction [i] would yield (145, 231, 638)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3493 770256 422529 887203 089822 056085 862785 655698 567789 155849 709555 478298 > 2²³¹ [i]