MinT - Best Known (137−91, 137, s)-Nets in Base 5

Best Known (137−91, 137, s)-Nets in Base 5

(137−91, 137, 78)-Net over F₅ — Constructive and digital

Digital (46, 137, 78)-net over F₅, using

t-expansion [i] based on digital (38, 137, 78)-net over F₅, using
- net from sequence [i] based on digital (38, 77)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 38 and N(F) ≥ 78, using
    - an explicitly constructive algebraic function field [i]

(137−91, 137, 88)-Net over F₅ — Digital

Digital (46, 137, 88)-net over F₅, using

t-expansion [i] based on digital (45, 137, 88)-net over F₅, using
- net from sequence [i] based on digital (45, 87)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 45 and N(F) ≥ 88, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(137−91, 137, 538)-Net in Base 5 — Upper bound on s

There is no (46, 137, 539)-net in base 5, because

1 times m-reduction [i] would yield (46, 136, 539)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 121163 380452 373241 526553 881033 150297 846395 661946 038066 978150 916685 697303 455190 900794 556265 940925 > 5¹³⁶ [i]