MinT - Best Known (58−30, 58, s)-Nets in Base 4

Best Known (58−30, 58, s)-Nets in Base 4

(58−30, 58, 41)-Net over F₄ — Constructive and digital

Digital (28, 58, 41)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 18, 14)-net over F₄, using
  - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]
2. digital (10, 40, 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]

(58−30, 58, 42)-Net in Base 4 — Constructive

(28, 58, 42)-net in base 4, using

t-expansion [i] based on (27, 58, 42)-net in base 4, using
- net from sequence [i] based on (27, 41)-sequence in base 4, using
  - base expansion [i] based on digital (54, 41)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
      - an explicitly constructive algebraic function field [i]

(58−30, 58, 55)-Net over F₄ — Digital

Digital (28, 58, 55)-net over F₄, using

t-expansion [i] based on digital (26, 58, 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]

(58−30, 58, 443)-Net in Base 4 — Upper bound on s

There is no (28, 58, 444)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 83764 379770 057526 040985 953646 100692 > 4⁵⁸ [i]