MinT - Best Known (57−29, 57, s)-Nets in Base 4

Best Known (57−29, 57, s)-Nets in Base 4

(57−29, 57, 42)-Net over F₄ — Constructive and digital

Digital (28, 57, 42)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (7, 21, 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]
2. digital (7, 36, 21)-net over F₄, using
  - net from sequence [i] based on digital (7, 20)-sequence over F₄ (see above)

(57−29, 57, 45)-Net in Base 4 — Constructive

(28, 57, 45)-net in base 4, using

base change [i] based on digital (9, 38, 45)-net over F₈, using
- net from sequence [i] based on digital (9, 44)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using
    - an explicitly constructive algebraic function field [i]

(57−29, 57, 55)-Net over F₄ — Digital

Digital (28, 57, 55)-net over F₄, using

t-expansion [i] based on digital (26, 57, 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]

(57−29, 57, 504)-Net in Base 4 — Upper bound on s

There is no (28, 57, 505)-net in base 4, because

1 times m-reduction [i] would yield (28, 56, 505)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5214 967335 728799 908533 523244 567736 > 4⁵⁶ [i]