MinT - Best Known (115, 115+41, s)-Nets in Base 4

Best Known (115, 115+41, s)-Nets in Base 4

(115, 115+41, 531)-Net over F₄ — Constructive and digital

Digital (115, 156, 531)-net over F₄, using

6 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+41, 576)-Net in Base 4 — Constructive

(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+41, 1191)-Net over F₄ — Digital

Digital (115, 156, 1191)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(115, 115+41, 128261)-Net in Base 4 — Upper bound on s

There is no (115, 156, 128262)-net in base 4, because

1 times m-reduction [i] would yield (115, 155, 128262)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2086 196981 469466 955142 815551 854531 029744 644531 112112 296603 869713 007043 922366 741581 625959 918896 > 4¹⁵⁵ [i]