MinT - Best Known (57, 57+22, s)-Nets in Base 5

Best Known (57, 57+22, s)-Nets in Base 5

(57, 57+22, 270)-Net over F₅ — Constructive and digital

Digital (57, 79, 270)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (4, 15, 18)-net over F₅, using
  - net from sequence [i] based on digital (4, 17)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 4 and N(F) ≥ 18, using
      - an explicitly constructive algebraic function field [i]
2. digital (42, 64, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 32, 126)-net over F₂₅, using
    - net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
        the Hermitian function field over F₂₅ [i]

(57, 57+22, 935)-Net over F₅ — Digital

Digital (57, 79, 935)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(57, 57+22, 128470)-Net in Base 5 — Upper bound on s

There is no (57, 79, 128471)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 16 544304 272984 389915 450295 329588 027687 484683 402202 878645 > 5⁷⁹ [i]