MinT - Best Known (129, 129+31, s)-Nets in Base 4

Best Known (129, 129+31, s)-Nets in Base 4

(129, 129+31, 1104)-Net over F₄ — Constructive and digital

Digital (129, 160, 1104)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (21, 36, 76)-net over F₄, using
  - trace code for nets [i] based on digital (3, 18, 38)-net over F₁₆, using
    - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
        a function field by SÃ©mirat [i]
2. digital (93, 124, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 31, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(129, 129+31, 6541)-Net over F₄ — Digital

Digital (129, 160, 6541)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(129, 129+31, 5157987)-Net in Base 4 — Upper bound on s

There is no (129, 160, 5157988)-net in base 4, because

1 times m-reduction [i] would yield (129, 159, 5157988)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 533997 565015 824774 318942 791574 907466 416560 247829 761671 346252 215334 453748 234767 453199 003923 226200 > 4¹⁵⁹ [i]