MinT - Best Known (103−29, 103, s)-Nets in Base 4

Best Known (103−29, 103, s)-Nets in Base 4

(103−29, 103, 384)-Net over F₄ — Constructive and digital

Digital (74, 103, 384)-net over F₄, using

4¹ times duplication [i] based on digital (73, 102, 384)-net over F₄, using
- trace code for nets [i] based on digital (5, 34, 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]

(103−29, 103, 387)-Net in Base 4 — Constructive

(74, 103, 387)-net in base 4, using

4¹ times duplication [i] based on (73, 102, 387)-net in base 4, using
- trace code for nets [i] based on (5, 34, 129)-net in base 64, using
  - 1 times m-reduction [i] based on (5, 35, 129)-net in base 64, using
    - base change [i] based on digital (0, 30, 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]

(103−29, 103, 631)-Net over F₄ — Digital

Digital (74, 103, 631)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(103−29, 103, 49054)-Net in Base 4 — Upper bound on s

There is no (74, 103, 49055)-net in base 4, because

1 times m-reduction [i] would yield (74, 102, 49055)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 25 718101 572624 356908 421843 580180 668798 801275 920132 190621 985131 > 4¹⁰² [i]