MinT - Best Known (27, 57, s)-Nets in Base 4

Best Known (27, 57, s)-Nets in Base 4

(27, 57, 38)-Net over F₄ — Constructive and digital

Digital (27, 57, 38)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 20, 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, 37, 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]

(27, 57, 42)-Net in Base 4 — Constructive

(27, 57, 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]

(27, 57, 55)-Net over F₄ — Digital

Digital (27, 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]

(27, 57, 403)-Net in Base 4 — Upper bound on s

There is no (27, 57, 404)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 21117 373391 656722 201348 760474 068604 > 4⁵⁷ [i]