MinT - Best Known (129−43, 129, s)-Nets in Base 4

Best Known (129−43, 129, s)-Nets in Base 4

(129−43, 129, 195)-Net over F₄ — Constructive and digital

Digital (86, 129, 195)-net over F₄, using

trace code for nets [i] based on digital (0, 43, 65)-net over F₆₄, using
- net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
  - generalized Faure sequence [i]
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
    - the rational function field F₆₄(x) [i]
  - Niederreiter sequence [i]

(129−43, 129, 196)-Net in Base 4 — Constructive

(86, 129, 196)-net in base 4, using

t-expansion [i] based on (85, 129, 196)-net in base 4, using
- 1 times m-reduction [i] based on (85, 130, 196)-net in base 4, using
  - trace code for nets [i] based on (20, 65, 98)-net in base 16, using
    - base change [i] based on digital (7, 52, 98)-net over F₃₂, using
      - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(129−43, 129, 381)-Net over F₄ — Digital

Digital (86, 129, 381)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(129−43, 129, 13505)-Net in Base 4 — Upper bound on s

There is no (86, 129, 13506)-net in base 4, because

1 times m-reduction [i] would yield (86, 128, 13506)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 115805 951918 190395 096092 704677 606575 612185 825699 019829 556820 241103 396448 245616 > 4¹²⁸ [i]