MinT - Best Known (69, 69+118, s)-Nets in Base 4

Best Known (69, 69+118, s)-Nets in Base 4

(69, 69+118, 66)-Net over F₄ — Constructive and digital

Digital (69, 187, 66)-net over F₄, using

t-expansion [i] based on digital (49, 187, 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, 69+118, 99)-Net over F₄ — Digital

Digital (69, 187, 99)-net over F₄, using

t-expansion [i] based on digital (61, 187, 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, 69+118, 568)-Net in Base 4 — Upper bound on s

There is no (69, 187, 569)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 40184 112405 633007 960584 030084 037226 956212 939916 872557 456208 773785 858987 526293 625827 767383 120254 805407 267124 566336 > 4¹⁸⁷ [i]