MinT - Best Known (115−55, 115, s)-Nets in Base 5

Best Known (115−55, 115, s)-Nets in Base 5

(115−55, 115, 84)-Net over F₅ — Constructive and digital

Digital (60, 115, 84)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (2, 29, 12)-net over F₅, using
  - net from sequence [i] based on digital (2, 11)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 2 and N(F) ≥ 12, using
      - an explicitly constructive algebraic function field [i]
2. digital (31, 86, 72)-net over F₅, using
  - net from sequence [i] based on digital (31, 71)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 31 and N(F) ≥ 72, using
      - an explicitly constructive algebraic function field [i]

(115−55, 115, 136)-Net over F₅ — Digital

Digital (60, 115, 136)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(115−55, 115, 2421)-Net in Base 5 — Upper bound on s

There is no (60, 115, 2422)-net in base 5, because

1 times m-reduction [i] would yield (60, 114, 2422)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 48 561822 852173 923722 970271 660386 309620 892872 895634 604658 183854 989931 641520 780025 > 5¹¹⁴ [i]