MinT - Best Known (69, 194, s)-Nets in Base 4

Best Known (69, 194, s)-Nets in Base 4

(69, 194, 66)-Net over F₄ — Constructive and digital

Digital (69, 194, 66)-net over F₄, using

t-expansion [i] based on digital (49, 194, 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]

(69, 194, 99)-Net over F₄ — Digital

Digital (69, 194, 99)-net over F₄, using

t-expansion [i] based on digital (61, 194, 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]

(69, 194, 547)-Net in Base 4 — Upper bound on s

There is no (69, 194, 548)-net in base 4, because

1 times m-reduction [i] would yield (69, 193, 548)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 165 764468 172542 002184 947901 065498 691756 568979 012626 263427 372465 645589 028692 578742 833945 316943 179846 170813 214828 950400 > 4¹⁹³ [i]