MinT - Best Known (210−70, 210, s)-Nets in Base 4

Best Known (210−70, 210, s)-Nets in Base 4

(210−70, 210, 195)-Net over F₄ — Constructive and digital

Digital (140, 210, 195)-net over F₄, using

trace code for nets [i] based on digital (0, 70, 65)-net over F₆₄, using
- net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
  - generalized Faure sequence [i]
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
    - the rational function field F₆₄(x) [i]
  - Niederreiter sequence [i]

(210−70, 210, 240)-Net in Base 4 — Constructive

(140, 210, 240)-net in base 4, using

4 times m-reduction [i] based on (140, 214, 240)-net in base 4, using
- trace code for nets [i] based on (33, 107, 120)-net in base 16, using
  - 3 times m-reduction [i] based on (33, 110, 120)-net in base 16, using
    - base change [i] based on digital (11, 88, 120)-net over F₃₂, using
      - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(210−70, 210, 574)-Net over F₄ — Digital

Digital (140, 210, 574)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(210−70, 210, 18960)-Net in Base 4 — Upper bound on s

There is no (140, 210, 18961)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 710846 533224 888276 354638 040395 000629 702038 048541 139081 771883 977576 196931 533887 322640 858103 156565 735010 060863 707626 697416 327424 > 4²¹⁰ [i]