MinT - Best Known (193−69, 193, s)-Nets in Base 2

Best Known (193−69, 193, s)-Nets in Base 2

(193−69, 193, 69)-Net over F₂ — Constructive and digital

Digital (124, 193, 69)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (39, 73, 33)-net over F₂, using
  - net from sequence [i] based on digital (39, 32)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 39 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (51, 120, 36)-net over F₂, using
  - net from sequence [i] based on digital (51, 35)-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 3 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]

(193−69, 193, 84)-Net in Base 2 — Constructive

(124, 193, 84)-net in base 2, using

1 times m-reduction [i] based on (124, 194, 84)-net in base 2, using
- trace code for nets [i] based on (27, 97, 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]

(193−69, 193, 115)-Net over F₂ — Digital

Digital (124, 193, 115)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(193−69, 193, 629)-Net in Base 2 — Upper bound on s

There is no (124, 193, 630)-net in base 2, because

1 times m-reduction [i] would yield (124, 192, 630)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 6458 320241 632243 539790 829459 368014 675807 102577 544522 015434 > 2¹⁹² [i]