MinT - Best Known (61, 61+25, s)-Nets in Base 5

Best Known (61, 61+25, s)-Nets in Base 5

(61, 61+25, 270)-Net over F₅ — Constructive and digital

Digital (61, 86, 270)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (4, 16, 18)-net over F₅, using
  - net from sequence [i] based on digital (4, 17)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 4 and N(F) ≥ 18, using
      - an explicitly constructive algebraic function field [i]
2. digital (45, 70, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 35, 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]

(61, 61+25, 795)-Net over F₅ — Digital

Digital (61, 86, 795)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(61, 61+25, 118115)-Net in Base 5 — Upper bound on s

There is no (61, 86, 118116)-net in base 5, because

1 times m-reduction [i] would yield (61, 85, 118116)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 258501 293563 314397 731605 109117 735488 781647 027048 776897 090305 > 5⁸⁵ [i]