MinT - Best Known (125−35, 125, s)-Nets in Base 4

Best Known (125−35, 125, s)-Nets in Base 4

(125−35, 125, 384)-Net over F₄ — Constructive and digital

Digital (90, 125, 384)-net over F₄, using

t-expansion [i] based on digital (89, 125, 384)-net over F₄, using
- 1 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]

(125−35, 125, 387)-Net in Base 4 — Constructive

(90, 125, 387)-net in base 4, using

1 times m-reduction [i] based on (90, 126, 387)-net in base 4, using
- trace code for nets [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]

(125−35, 125, 754)-Net over F₄ — Digital

Digital (90, 125, 754)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(125−35, 125, 58914)-Net in Base 4 — Upper bound on s

There is no (90, 125, 58915)-net in base 4, because

1 times m-reduction [i] would yield (90, 124, 58915)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 452 423455 037482 343511 250998 921239 381728 813053 816679 645748 947950 728553 211900 > 4¹²⁴ [i]