MinT - Best Known (51−26, 51, s)-Nets in Base 4

Best Known (51−26, 51, s)-Nets in Base 4

(51−26, 51, 38)-Net over F₄ — Constructive and digital

Digital (25, 51, 38)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 18, 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, 33, 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]

(51−26, 51, 51)-Net over F₄ — Digital

Digital (25, 51, 51)-net over F₄, using

net from sequence [i] based on digital (25, 50)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 25 and N(F) ≥ 51, using
  - Drinfeld modules of rank 1 [i]

(51−26, 51, 424)-Net in Base 4 — Upper bound on s

There is no (25, 51, 425)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5 149362 986670 564481 348701 681856 > 4⁵¹ [i]