MinT - Best Known (68, 193, s)-Nets in Base 4

Best Known (68, 193, s)-Nets in Base 4

(68, 193, 66)-Net over F₄ — Constructive and digital

Digital (68, 193, 66)-net over F₄, using

t-expansion [i] based on digital (49, 193, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(68, 193, 99)-Net over F₄ — Digital

Digital (68, 193, 99)-net over F₄, using

t-expansion [i] based on digital (61, 193, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

(68, 193, 534)-Net in Base 4 — Upper bound on s

There is no (68, 193, 535)-net in base 4, because

1 times m-reduction [i] would yield (68, 192, 535)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 42 234851 318848 396734 072924 243038 139287 748030 852024 249404 557900 233776 517822 253468 478935 909572 752080 852120 926231 906880 > 4¹⁹² [i]