MinT - Best Known (89, 89+35, s)-Nets in Base 4

Best Known (89, 89+35, s)-Nets in Base 4

(89, 89+35, 384)-Net over F₄ — Constructive and digital

Digital (89, 124, 384)-net over F₄, using

2 times m-reduction [i] based on digital (89, 126, 384)-net over F₄, using
- trace code for nets [i] based on digital (5, 42, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(89, 89+35, 387)-Net in Base 4 — Constructive

(89, 124, 387)-net in base 4, using

4¹ times duplication [i] based on (88, 123, 387)-net in base 4, using
- trace code for nets [i] based on (6, 41, 129)-net in base 64, using
  - 1 times m-reduction [i] based on (6, 42, 129)-net in base 64, using
    - base change [i] based on digital (0, 36, 129)-net over F₁₂₈, using
      - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
        Niederreiter sequence [i]

(89, 89+35, 725)-Net over F₄ — Digital

Digital (89, 124, 725)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 89+35, 54299)-Net in Base 4 — Upper bound on s

There is no (89, 124, 54300)-net in base 4, because

1 times m-reduction [i] would yield (89, 123, 54300)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 113 096123 225204 267503 697182 567362 950021 364620 324786 156148 258886 691145 727891 > 4¹²³ [i]