MinT - Best Known (54, 54+19, s)-Nets in Base 5

Best Known (54, 54+19, s)-Nets in Base 5

(54, 54+19, 273)-Net over F₅ — Constructive and digital

Digital (54, 73, 273)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (6, 15, 21)-net over F₅, using
  - net from sequence [i] based on digital (6, 20)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 6 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]
2. digital (39, 58, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 29, 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]

(54, 54+19, 1299)-Net over F₅ — Digital

Digital (54, 73, 1299)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 54+19, 404990)-Net in Base 5 — Upper bound on s

There is no (54, 73, 404991)-net in base 5, because

1 times m-reduction [i] would yield (54, 72, 404991)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 211 760794 322528 361125 200132 010888 566012 825545 390045 > 5⁷² [i]