MinT - Best Known (245−64, 245, s)-Nets in Base 4

Best Known (245−64, 245, s)-Nets in Base 4

(245−64, 245, 536)-Net over F₄ — Constructive and digital

Digital (181, 245, 536)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 32, 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 (149, 213, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 71, 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]

(245−64, 245, 648)-Net in Base 4 — Constructive

(181, 245, 648)-net in base 4, using

1 times m-reduction [i] based on (181, 246, 648)-net in base 4, using
- trace code for nets [i] based on (17, 82, 216)-net in base 64, using
  - 2 times m-reduction [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]

(245−64, 245, 1809)-Net over F₄ — Digital

Digital (181, 245, 1809)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(245−64, 245, 173466)-Net in Base 4 — Upper bound on s

There is no (181, 245, 173467)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3197 011885 793892 866720 221715 504429 363415 224687 996316 631532 082406 675540 187979 187825 441519 872885 929860 493576 794325 314488 388930 481239 699804 188891 544300 > 4²⁴⁵ [i]