MinT - Best Known (56, 77, s)-Nets in Base 5

Best Known (56, 77, s)-Nets in Base 5

(56, 77, 272)-Net over F₅ — Constructive and digital

Digital (56, 77, 272)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (5, 15, 20)-net over F₅, using
  - net from sequence [i] based on digital (5, 19)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 5 and N(F) ≥ 20, using
      - an explicitly constructive algebraic function field [i]
2. digital (41, 62, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 31, 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]

(56, 77, 1029)-Net over F₅ — Digital

Digital (56, 77, 1029)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(56, 77, 232313)-Net in Base 5 — Upper bound on s

There is no (56, 77, 232314)-net in base 5, because

1 times m-reduction [i] would yield (56, 76, 232314)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 132349 273870 921662 644745 589408 822367 317836 675070 216001 > 5⁷⁶ [i]