MinT - Best Known (78−22, 78, s)-Nets in Base 5

Best Known (78−22, 78, s)-Nets in Base 5

(78−22, 78, 268)-Net over F₅ — Constructive and digital

Digital (56, 78, 268)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (3, 14, 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 (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]

(78−22, 78, 867)-Net over F₅ — Digital

Digital (56, 78, 867)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(78−22, 78, 110983)-Net in Base 5 — Upper bound on s

There is no (56, 78, 110984)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 3 309038 852201 427140 416220 844528 656095 668164 688140 512545 > 5⁷⁸ [i]