MinT - Best Known (55−29, 55, s)-Nets in Base 4

Best Known (55−29, 55, s)-Nets in Base 4

(55−29, 55, 38)-Net over F₄ — Constructive and digital

Digital (26, 55, 38)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 19, 17)-net over F₄, using
  - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [i]
2. digital (7, 36, 21)-net over F₄, using
  - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]

(55−29, 55, 55)-Net over F₄ — Digital

Digital (26, 55, 55)-net over F₄, using

net from sequence [i] based on digital (26, 54)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 26 and N(F) ≥ 55, using
  - Drinfeld modules of rank 1 [i]

(55−29, 55, 412)-Net in Base 4 — Upper bound on s

There is no (26, 55, 413)-net in base 4, because

1 times m-reduction [i] would yield (26, 54, 413)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 333 564425 521008 548668 460309 183368 > 4⁵⁴ [i]