MinT - Best Known (255−67, 255, s)-Nets in Base 4

Best Known (255−67, 255, s)-Nets in Base 4

(255−67, 255, 536)-Net over F₄ — Constructive and digital

Digital (188, 255, 536)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 33, 5)-net over F₄, using
  - net from sequence [i] based on digital (0, 4)-sequence over F₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 0 and N(F) ≥ 5, using
      - the rational function field F₄(x) [i]
    - Niederreiter sequence [i]
2. digital (155, 222, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 74, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(255−67, 255, 648)-Net in Base 4 — Constructive

(188, 255, 648)-net in base 4, using

4³ times duplication [i] based on (185, 252, 648)-net in base 4, using
- trace code for nets [i] based on (17, 84, 216)-net in base 64, using
  - base change [i] based on digital (5, 72, 216)-net over F₁₂₈, using
    - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(255−67, 255, 1828)-Net over F₄ — Digital

Digital (188, 255, 1828)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(255−67, 255, 188889)-Net in Base 4 — Upper bound on s

There is no (188, 255, 188890)-net in base 4, because

1 times m-reduction [i] would yield (188, 254, 188890)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 838 030718 989242 535750 939465 518113 559547 491025 049618 854395 214188 025343 929360 941352 525533 599726 345585 366731 469176 404580 822122 321761 689628 246015 262209 777050 > 4²⁵⁴ [i]