MinT - Best Known (54, 54+20, s)-Nets in Base 4

Best Known (54, 54+20, s)-Nets in Base 4

(54, 54+20, 312)-Net over F₄ — Constructive and digital

Digital (54, 74, 312)-net over F₄, using

t-expansion [i] based on digital (53, 74, 312)-net over F₄, using
- 1 times m-reduction [i] based on digital (53, 75, 312)-net over F₄, using
  - trace code for nets [i] based on digital (3, 25, 104)-net over F₆₄, using
    - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(54, 54+20, 387)-Net in Base 4 — Constructive

(54, 74, 387)-net in base 4, using

1 times m-reduction [i] based on (54, 75, 387)-net in base 4, using
- trace code for nets [i] based on (4, 25, 129)-net in base 64, using
  - 3 times m-reduction [i] based on (4, 28, 129)-net in base 64, using
    - base change [i] based on digital (0, 24, 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]

(54, 54+20, 594)-Net over F₄ — Digital

Digital (54, 74, 594)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 54+20, 43054)-Net in Base 4 — Upper bound on s

There is no (54, 74, 43055)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 356 841591 458623 587182 124685 691703 075885 549181 > 4⁷⁴ [i]