MinT - Best Known (50, 68, s)-Nets in Base 5

Best Known (50, 68, s)-Nets in Base 5

(50, 68, 268)-Net over F₅ — Constructive and digital

Digital (50, 68, 268)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (3, 12, 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 (38, 56, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 28, 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]

(50, 68, 1130)-Net over F₅ — Digital

Digital (50, 68, 1130)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(50, 68, 198054)-Net in Base 5 — Upper bound on s

There is no (50, 68, 198055)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 338822 722929 312323 922390 495825 916811 331716 400765 > 5⁶⁸ [i]