MinT - Best Known (194−111, 194, s)-Nets in Base 4

Best Known (194−111, 194, s)-Nets in Base 4

(194−111, 194, 104)-Net over F₄ — Constructive and digital

Digital (83, 194, 104)-net over F₄, using

t-expansion [i] based on digital (73, 194, 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]

(194−111, 194, 129)-Net over F₄ — Digital

Digital (83, 194, 129)-net over F₄, using

t-expansion [i] based on digital (81, 194, 129)-net over F₄, using
- net from sequence [i] based on digital (81, 128)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 81 and N(F) ≥ 129, using
    - Drinfeld modules of rank 1 [i]

(194−111, 194, 877)-Net in Base 4 — Upper bound on s

There is no (83, 194, 878)-net in base 4, because

1 times m-reduction [i] would yield (83, 193, 878)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 162 637076 857125 514498 615090 359334 733114 977906 074404 133875 547193 890600 645798 166635 369559 186171 555089 985243 380003 174544 > 4¹⁹³ [i]