MinT - Best Known (115, 115+39, s)-Nets in Base 4

Best Known (115, 115+39, s)-Nets in Base 4

(115, 115+39, 531)-Net over F₄ — Constructive and digital

Digital (115, 154, 531)-net over F₄, using

8 times m-reduction [i] based on digital (115, 162, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 54, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(115, 115+39, 576)-Net in Base 4 — Constructive

(115, 154, 576)-net in base 4, using

2 times m-reduction [i] based on (115, 156, 576)-net in base 4, using
- trace code for nets [i] based on (11, 52, 192)-net in base 64, using
  - 4 times m-reduction [i] based on (11, 56, 192)-net in base 64, using
    - base change [i] based on digital (3, 48, 192)-net over F₁₂₈, using
      - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(115, 115+39, 1398)-Net over F₄ — Digital

Digital (115, 154, 1398)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(115, 115+39, 186305)-Net in Base 4 — Upper bound on s

There is no (115, 154, 186306)-net in base 4, because

1 times m-reduction [i] would yield (115, 153, 186306)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 130 376188 741490 982727 330339 194655 027352 591200 037187 313869 155944 361363 481472 960909 529904 631648 > 4¹⁵³ [i]