MinT - Best Known (227−59, 227, s)-Nets in Base 4

Best Known (227−59, 227, s)-Nets in Base 4

(227−59, 227, 536)-Net over F₄ — Constructive and digital

Digital (168, 227, 536)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 29, 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 (139, 198, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 66, 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]

(227−59, 227, 648)-Net in Base 4 — Constructive

(168, 227, 648)-net in base 4, using

1 times m-reduction [i] based on (168, 228, 648)-net in base 4, using
- trace code for nets [i] based on (16, 76, 216)-net in base 64, using
  - 1 times m-reduction [i] based on (16, 77, 216)-net in base 64, using
    - base change [i] based on digital (5, 66, 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]

(227−59, 227, 1729)-Net over F₄ — Digital

Digital (168, 227, 1729)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(227−59, 227, 191365)-Net in Base 4 — Upper bound on s

There is no (168, 227, 191366)-net in base 4, because

1 times m-reduction [i] would yield (168, 226, 191366)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11630 907130 277186 136762 228897 640368 492860 544171 055307 392886 346888 048492 079773 813398 085082 493630 680976 933965 866935 691038 475278 027859 418096 > 4²²⁶ [i]