MinT - Best Known (68, 68+71, s)-Nets in Base 5

Best Known (68, 68+71, s)-Nets in Base 5

(68, 68+71, 84)-Net over F₅ — Constructive and digital

Digital (68, 139, 84)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (2, 37, 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, 102, 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]

(68, 68+71, 132)-Net over F₅ — Digital

Digital (68, 139, 132)-net over F₅, using

t-expansion [i] based on digital (67, 139, 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]

(68, 68+71, 1956)-Net in Base 5 — Upper bound on s

There is no (68, 139, 1957)-net in base 5, because

1 times m-reduction [i] would yield (68, 138, 1957)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 2 892912 595409 840655 857714 523123 407023 278207 937852 392158 114055 444612 730999 223883 988010 922654 286285 > 5¹³⁸ [i]