MinT - Best Known (168−33, 168, s)-Nets in Base 4

Best Known (168−33, 168, s)-Nets in Base 4

(168−33, 168, 1094)-Net over F₄ — Constructive and digital

Digital (135, 168, 1094)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (20, 36, 66)-net over F₄, using
  - trace code for nets [i] based on digital (2, 18, 33)-net over F₁₆, using
    - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
        a function field by SÃ©mirat [i]
2. digital (99, 132, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 33, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(168−33, 168, 6190)-Net over F₄ — Digital

Digital (135, 168, 6190)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(168−33, 168, 4359305)-Net in Base 4 — Upper bound on s

There is no (135, 168, 4359306)-net in base 4, because

1 times m-reduction [i] would yield (135, 167, 4359306)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 34996 017240 066321 548107 222846 887882 933591 559164 530553 497548 552909 103613 108265 167540 544990 659210 428392 > 4¹⁶⁷ [i]