MinT - Best Known (193−117, 193, s)-Nets in Base 4

Best Known (193−117, 193, s)-Nets in Base 4

(193−117, 193, 104)-Net over F₄ — Constructive and digital

Digital (76, 193, 104)-net over F₄, using

t-expansion [i] based on digital (73, 193, 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]

(193−117, 193, 112)-Net over F₄ — Digital

Digital (76, 193, 112)-net over F₄, using

t-expansion [i] based on digital (73, 193, 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]

(193−117, 193, 689)-Net in Base 4 — Upper bound on s

There is no (76, 193, 690)-net in base 4, because

1 times m-reduction [i] would yield (76, 192, 690)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 40 199899 237550 361359 241188 427343 758008 665013 005466 323109 455405 763974 095595 582905 110588 231240 667560 390548 773298 569280 > 4¹⁹² [i]