MinT - Best Known (91−45, 91, s)-Nets in Base 4

Best Known (91−45, 91, s)-Nets in Base 4

(91−45, 91, 57)-Net over F₄ — Constructive and digital

Digital (46, 91, 57)-net over F₄, using

1 times m-reduction [i] based on digital (46, 92, 57)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 33, 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, 59, 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]

(91−45, 91, 65)-Net in Base 4 — Constructive

(46, 91, 65)-net in base 4, using

5 times m-reduction [i] based on (46, 96, 65)-net in base 4, using
- base change [i] based on digital (14, 64, 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]

(91−45, 91, 81)-Net over F₄ — Digital

Digital (46, 91, 81)-net over F₄, using

net from sequence [i] based on digital (46, 80)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 46 and N(F) ≥ 81, using
  - Drinfeld modules of rank 1 [i]

(91−45, 91, 858)-Net in Base 4 — Upper bound on s

There is no (46, 91, 859)-net in base 4, because

1 times m-reduction [i] would yield (46, 90, 859)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 543709 951242 956091 430461 587675 443323 421014 737112 709230 > 4⁹⁰ [i]