MinT - Best Known (196−51, 196, s)-Nets in Base 4

Best Known (196−51, 196, s)-Nets in Base 4

(196−51, 196, 531)-Net over F₄ — Constructive and digital

Digital (145, 196, 531)-net over F₄, using

11 times m-reduction [i] based on digital (145, 207, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 69, 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]

(196−51, 196, 576)-Net in Base 4 — Constructive

(145, 196, 576)-net in base 4, using

2 times m-reduction [i] based on (145, 198, 576)-net in base 4, using
- trace code for nets [i] based on (13, 66, 192)-net in base 64, using
  - 4 times m-reduction [i] based on (13, 70, 192)-net in base 64, using
    - base change [i] based on digital (3, 60, 192)-net over F₁₂₈, using
      - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(196−51, 196, 1513)-Net over F₄ — Digital

Digital (145, 196, 1513)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(196−51, 196, 168469)-Net in Base 4 — Upper bound on s

There is no (145, 196, 168470)-net in base 4, because

1 times m-reduction [i] would yield (145, 195, 168470)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2522 088751 381473 068257 560866 443094 237421 950650 589346 854792 879027 414763 153693 090503 681167 284565 320622 546251 557167 229248 > 4¹⁹⁵ [i]