MinT - Best Known (123, 123+65, s)-Nets in Base 2

Best Known (123, 123+65, s)-Nets in Base 2

(123, 123+65, 70)-Net over F₂ — Constructive and digital

Digital (123, 188, 70)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (21, 53, 21)-net over F₂, using
  - net from sequence [i] based on digital (21, 20)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 21 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]
2. digital (70, 135, 49)-net over F₂, using
  - net from sequence [i] based on digital (70, 48)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, and 1 place with degree 2 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(123, 123+65, 84)-Net in Base 2 — Constructive

(123, 188, 84)-net in base 2, using

4 times m-reduction [i] based on (123, 192, 84)-net in base 2, using
- trace code for nets [i] based on (27, 96, 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]

(123, 123+65, 122)-Net over F₂ — Digital

Digital (123, 188, 122)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(123, 123+65, 688)-Net in Base 2 — Upper bound on s

There is no (123, 188, 689)-net in base 2, because

1 times m-reduction [i] would yield (123, 187, 689)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 201 257533 370410 670086 959828 395189 095003 970487 541641 353859 > 2¹⁸⁷ [i]