MinT - Best Known (85−41, 85, s)-Nets in Base 4

Best Known (85−41, 85, s)-Nets in Base 4

(85−41, 85, 57)-Net over F₄ — Constructive and digital

Digital (44, 85, 57)-net over F₄, using

1 times m-reduction [i] based on digital (44, 86, 57)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 31, 27)-net over F₄, using
    - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
        an explicitly constructive algebraic function field [i]
  2. digital (13, 55, 30)-net over F₄, using
    - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
        F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(85−41, 85, 65)-Net in Base 4 — Constructive

(44, 85, 65)-net in base 4, using

5 times m-reduction [i] based on (44, 90, 65)-net in base 4, using
- base change [i] based on digital (14, 60, 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]

(85−41, 85, 80)-Net over F₄ — Digital

Digital (44, 85, 80)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(85−41, 85, 918)-Net in Base 4 — Upper bound on s

There is no (44, 85, 919)-net in base 4, because

1 times m-reduction [i] would yield (44, 84, 919)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 374 184404 858236 128181 944069 536278 850834 983458 257839 > 4⁸⁴ [i]