MinT - Best Known (52−26, 52, s)-Nets in Base 4

Best Known (52−26, 52, s)-Nets in Base 4

(52−26, 52, 41)-Net over F₄ — Constructive and digital

Digital (26, 52, 41)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 16, 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, 36, 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]

(52−26, 52, 55)-Net over F₄ — Digital

Digital (26, 52, 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]

(52−26, 52, 473)-Net in Base 4 — Upper bound on s

There is no (26, 52, 474)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 20 614837 558881 481275 555635 512132 > 4⁵² [i]