MinT - Best Known (59, 59+24, s)-Nets in Base 5

Best Known (59, 59+24, s)-Nets in Base 5

(59, 59+24, 268)-Net over F₅ — Constructive and digital

Digital (59, 83, 268)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (3, 15, 16)-net over F₅, using
  - net from sequence [i] based on digital (3, 15)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 3 and N(F) ≥ 16, using
      - an explicitly constructive algebraic function field [i]
2. digital (44, 68, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 34, 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]

(59, 59+24, 796)-Net over F₅ — Digital

Digital (59, 83, 796)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(59, 59+24, 90323)-Net in Base 5 — Upper bound on s

There is no (59, 83, 90324)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 10339 779363 829257 279406 863511 488556 972561 007441 880247 343873 > 5⁸³ [i]