MinT - Best Known (133, 133+45, s)-Nets in Base 4

Best Known (133, 133+45, s)-Nets in Base 4

(133, 133+45, 536)-Net over F₄ — Constructive and digital

Digital (133, 178, 536)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 22, 5)-net over F₄, using
  - net from sequence [i] based on digital (0, 4)-sequence over F₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 0 and N(F) ≥ 5, using
      - the rational function field F₄(x) [i]
    - Niederreiter sequence [i]
2. digital (111, 156, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 52, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(133, 133+45, 648)-Net in Base 4 — Constructive

(133, 178, 648)-net in base 4, using

4¹ times duplication [i] based on (132, 177, 648)-net in base 4, using
- trace code for nets [i] based on (14, 59, 216)-net in base 64, using
  - 4 times m-reduction [i] based on (14, 63, 216)-net in base 64, using
    - base change [i] based on digital (5, 54, 216)-net over F₁₂₈, using
      - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(133, 133+45, 1590)-Net over F₄ — Digital

Digital (133, 178, 1590)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(133, 133+45, 210640)-Net in Base 4 — Upper bound on s

There is no (133, 178, 210641)-net in base 4, because

1 times m-reduction [i] would yield (133, 177, 210641)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 36698 965937 463253 441006 622961 447912 333971 791761 578385 359522 103850 226189 659106 174872 699831 820977 349675 049184 > 4¹⁷⁷ [i]