MinT - Best Known (199−66, 199, s)-Nets in Base 4

Best Known (199−66, 199, s)-Nets in Base 4

(199−66, 199, 195)-Net over F₄ — Constructive and digital

Digital (133, 199, 195)-net over F₄, using

4¹ times duplication [i] based on digital (132, 198, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 66, 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]

(199−66, 199, 240)-Net in Base 4 — Constructive

(133, 199, 240)-net in base 4, using

t-expansion [i] based on (131, 199, 240)-net in base 4, using
- 1 times m-reduction [i] based on (131, 200, 240)-net in base 4, using
  - trace code for nets [i] based on (31, 100, 120)-net in base 16, using
    - base change [i] based on digital (11, 80, 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]

(199−66, 199, 558)-Net over F₄ — Digital

Digital (133, 199, 558)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(199−66, 199, 18715)-Net in Base 4 — Upper bound on s

There is no (133, 199, 18716)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 645646 462242 107098 115035 509734 006785 385078 699695 453423 213375 166466 051535 621759 068545 472716 323098 869155 175958 453849 839656 > 4¹⁹⁹ [i]