MinT - Best Known (227−75, 227, s)-Nets in Base 4

Best Known (227−75, 227, s)-Nets in Base 4

(227−75, 227, 195)-Net over F₄ — Constructive and digital

Digital (152, 227, 195)-net over F₄, using

1 times m-reduction [i] based on digital (152, 228, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 76, 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]

(227−75, 227, 240)-Net in Base 4 — Constructive

(152, 227, 240)-net in base 4, using

7 times m-reduction [i] based on (152, 234, 240)-net in base 4, using
- trace code for nets [i] based on (35, 117, 120)-net in base 16, using
  - 3 times m-reduction [i] based on (35, 120, 120)-net in base 16, using
    - base change [i] based on digital (11, 96, 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]

(227−75, 227, 635)-Net over F₄ — Digital

Digital (152, 227, 635)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(227−75, 227, 23209)-Net in Base 4 — Upper bound on s

There is no (152, 227, 23210)-net in base 4, because

1 times m-reduction [i] would yield (152, 226, 23210)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11633 385244 641848 352526 375110 912925 025588 235660 208661 422657 908317 703382 496921 216644 804183 024408 776997 395554 218817 101599 391628 982487 236370 > 4²²⁶ [i]