MinT - Best Known (69, 69+128, s)-Nets in Base 4

Best Known (69, 69+128, s)-Nets in Base 4

(69, 69+128, 66)-Net over F₄ — Constructive and digital

Digital (69, 197, 66)-net over F₄, using

t-expansion [i] based on digital (49, 197, 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+128, 99)-Net over F₄ — Digital

Digital (69, 197, 99)-net over F₄, using

t-expansion [i] based on digital (61, 197, 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+128, 535)-Net in Base 4 — Upper bound on s

There is no (69, 197, 536)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 42645 229753 454959 302351 732772 506686 334814 745647 538898 479638 700483 193650 078519 119126 255960 388275 126539 807319 313865 227800 > 4¹⁹⁷ [i]