MinT - Best Known (75, 75+115, s)-Nets in Base 4

Best Known (75, 75+115, s)-Nets in Base 4

(75, 75+115, 104)-Net over F₄ — Constructive and digital

Digital (75, 190, 104)-net over F₄, using

t-expansion [i] based on digital (73, 190, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(75, 75+115, 112)-Net over F₄ — Digital

Digital (75, 190, 112)-net over F₄, using

t-expansion [i] based on digital (73, 190, 112)-net over F₄, using
- net from sequence [i] based on digital (73, 111)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 112, using
    - Drinfeld modules of rank 1 [i]

(75, 75+115, 683)-Net in Base 4 — Upper bound on s

There is no (75, 190, 684)-net in base 4, because

1 times m-reduction [i] would yield (75, 189, 684)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 621637 791048 390978 536055 176999 202714 227429 941734 352657 910023 617955 563552 710859 852045 937903 120168 945289 638152 429880 > 4¹⁸⁹ [i]