MinT - Best Known (135−68, 135, s)-Nets in Base 5

Best Known (135−68, 135, s)-Nets in Base 5

(135−68, 135, 84)-Net over F₅ — Constructive and digital

Digital (67, 135, 84)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (2, 36, 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, 99, 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]

(135−68, 135, 132)-Net over F₅ — Digital

Digital (67, 135, 132)-net over F₅, using

net from sequence [i] based on digital (67, 131)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 67 and N(F) ≥ 132, using
  - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(135−68, 135, 1992)-Net in Base 5 — Upper bound on s

There is no (67, 135, 1993)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 23248 201001 493853 785237 751424 661029 658496 396956 576478 617753 394465 269216 858359 222761 866967 783065 > 5¹³⁵ [i]