MinT - Best Known (194−117, 194, s)-Nets in Base 4

Best Known (194−117, 194, s)-Nets in Base 4

(194−117, 194, 104)-Net over F₄ — Constructive and digital

Digital (77, 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]

(194−117, 194, 112)-Net over F₄ — Digital

Digital (77, 194, 112)-net over F₄, using

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

(194−117, 194, 707)-Net in Base 4 — Upper bound on s

There is no (77, 194, 708)-net in base 4, because

1 times m-reduction [i] would yield (77, 193, 708)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 163 371442 077376 693477 135907 907330 131964 829236 434043 919581 707790 878056 371603 930377 491090 885699 712996 146385 404887 004600 > 4¹⁹³ [i]