MinT - Best Known (226−75, 226, s)-Nets in Base 4

Best Known (226−75, 226, s)-Nets in Base 4

(226−75, 226, 195)-Net over F₄ — Constructive and digital

Digital (151, 226, 195)-net over F₄, using

4¹ times duplication [i] based on digital (150, 225, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 75, 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]

(226−75, 226, 240)-Net in Base 4 — Constructive

(151, 226, 240)-net in base 4, using

t-expansion [i] based on (149, 226, 240)-net in base 4, using
- 4 times m-reduction [i] based on (149, 230, 240)-net in base 4, using
  - trace code for nets [i] based on (34, 115, 120)-net in base 16, using
    - base change [i] based on digital (11, 92, 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]

(226−75, 226, 622)-Net over F₄ — Digital

Digital (151, 226, 622)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(226−75, 226, 22355)-Net in Base 4 — Upper bound on s

There is no (151, 226, 22356)-net in base 4, because

1 times m-reduction [i] would yield (151, 225, 22356)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2911 440962 076350 009719 864220 200922 508089 733596 796659 315794 931376 865478 033808 724123 447273 261250 078832 972084 346324 023053 622014 654761 677400 > 4²²⁵ [i]