MinT - Best Known (162−52, 162, s)-Nets in Base 2

Best Known (162−52, 162, s)-Nets in Base 2

(162−52, 162, 72)-Net over F₂ — Constructive and digital

Digital (110, 162, 72)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (2, 28, 6)-net over F₂, using
  - net from sequence [i] based on digital (2, 5)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 2 and N(F) ≥ 6, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (82, 134, 66)-net over F₂, using
  - trace code for nets [i] based on digital (15, 67, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]

(162−52, 162, 84)-Net in Base 2 — Constructive

(110, 162, 84)-net in base 2, using

4 times m-reduction [i] based on (110, 166, 84)-net in base 2, using
- trace code for nets [i] based on (27, 83, 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]

(162−52, 162, 128)-Net over F₂ — Digital

Digital (110, 162, 128)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(162−52, 162, 754)-Net in Base 2 — Upper bound on s

There is no (110, 162, 755)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 5 898725 894525 614145 388000 397230 502470 134947 909172 > 2¹⁶² [i]