MinT - Best Known (68, 68+69, s)-Nets in Base 5

Best Known (68, 68+69, s)-Nets in Base 5

(68, 68+69, 88)-Net over F₅ — Constructive and digital

Digital (68, 137, 88)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (3, 37, 16)-net over F₅, using
  - net from sequence [i] based on digital (3, 15)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 3 and N(F) ≥ 16, using
      - an explicitly constructive algebraic function field [i]
2. digital (31, 100, 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+69, 132)-Net over F₅ — Digital

Digital (68, 137, 132)-net over F₅, using

t-expansion [i] based on digital (67, 137, 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+69, 2090)-Net in Base 5 — Upper bound on s

There is no (68, 137, 2091)-net in base 5, because

1 times m-reduction [i] would yield (68, 136, 2091)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 116593 693630 737415 253542 415926 703213 531133 362779 003657 593980 440388 293850 507340 611202 531356 598265 > 5¹³⁶ [i]