MinT - Best Known (58, 58+21, s)-Nets in Base 5

Best Known (58, 58+21, s)-Nets in Base 5

(58, 58+21, 274)-Net over F₅ — Constructive and digital

Digital (58, 79, 274)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (7, 17, 22)-net over F₅, using
  - net from sequence [i] based on digital (7, 21)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 7 and N(F) ≥ 22, 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]

(58, 58+21, 1207)-Net over F₅ — Digital

Digital (58, 79, 1207)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(58, 58+21, 320533)-Net in Base 5 — Upper bound on s

There is no (58, 79, 320534)-net in base 5, because

1 times m-reduction [i] would yield (58, 78, 320534)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 3 308817 045540 041584 828721 258414 682597 009911 025709 854465 > 5⁷⁸ [i]