MinT - Best Known (86−43, 86, s)-Nets in Base 4

Best Known (86−43, 86, s)-Nets in Base 4

(86−43, 86, 56)-Net over F₄ — Constructive and digital

Digital (43, 86, 56)-net over F₄, using

t-expansion [i] based on digital (33, 86, 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]

(86−43, 86, 65)-Net in Base 4 — Constructive

(43, 86, 65)-net in base 4, using

1 times m-reduction [i] based on (43, 87, 65)-net in base 4, using
- base change [i] based on digital (14, 58, 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]

(86−43, 86, 75)-Net over F₄ — Digital

Digital (43, 86, 75)-net over F₄, using

t-expansion [i] based on digital (40, 86, 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]

(86−43, 86, 774)-Net in Base 4 — Upper bound on s

There is no (43, 86, 775)-net in base 4, because

1 times m-reduction [i] would yield (43, 85, 775)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1523 438677 249213 706158 445045 622630 723098 783626 326496 > 4⁸⁵ [i]