MinT - Best Known (84−42, 84, s)-Nets in Base 4

Best Known (84−42, 84, s)-Nets in Base 4

(84−42, 84, 56)-Net over F₄ — Constructive and digital

Digital (42, 84, 56)-net over F₄, using

t-expansion [i] based on digital (33, 84, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
    - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(84−42, 84, 65)-Net in Base 4 — Constructive

(42, 84, 65)-net in base 4, using

base change [i] based on digital (14, 56, 65)-net over F₈, using
- net from sequence [i] based on digital (14, 64)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
    - the Suzuki function field over F₈ [i]

(84−42, 84, 75)-Net over F₄ — Digital

Digital (42, 84, 75)-net over F₄, using

t-expansion [i] based on digital (40, 84, 75)-net over F₄, using
- net from sequence [i] based on digital (40, 74)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 40 and N(F) ≥ 75, using
    - Drinfeld modules of rank 1 [i]

(84−42, 84, 723)-Net in Base 4 — Upper bound on s

There is no (42, 84, 724)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 376 314640 588221 636476 709913 654972 525180 704281 954128 > 4⁸⁴ [i]