MinT - Best Known (58−32, 58, s)-Nets in Base 4

Best Known (58−32, 58, s)-Nets in Base 4

(58−32, 58, 35)-Net over F₄ — Constructive and digital

Digital (26, 58, 35)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 19, 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 (7, 39, 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]

(58−32, 58, 36)-Net in Base 4 — Constructive

(26, 58, 36)-net in base 4, using

net from sequence [i] based on (26, 35)-sequence in base 4, using
- base expansion [i] based on digital (52, 35)-sequence over F₂, using
  - t-expansion [i] based on digital (51, 35)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 3 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

(58−32, 58, 55)-Net over F₄ — Digital

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]

(58−32, 58, 332)-Net in Base 4 — Upper bound on s

There is no (26, 58, 333)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 85415 914873 767830 645931 939292 402900 > 4⁵⁸ [i]