MinT - Best Known (133−83, 133, s)-Nets in Base 5

Best Known (133−83, 133, s)-Nets in Base 5

(133−83, 133, 82)-Net over F₅ — Constructive and digital

Digital (50, 133, 82)-net over F₅, using

t-expansion [i] based on digital (48, 133, 82)-net over F₅, using
- net from sequence [i] based on digital (48, 81)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 48 and N(F) ≥ 82, using
    - an explicitly constructive algebraic function field [i]

(133−83, 133, 96)-Net over F₅ — Digital

Digital (50, 133, 96)-net over F₅, using

t-expansion [i] based on digital (49, 133, 96)-net over F₅, using
- net from sequence [i] based on digital (49, 95)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 49 and N(F) ≥ 96, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(133−83, 133, 688)-Net in Base 5 — Upper bound on s

There is no (50, 133, 689)-net in base 5, because

1 times m-reduction [i] would yield (50, 132, 689)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 191 027573 745313 438586 627464 428335 794761 723876 631692 331889 945244 753814 365547 235556 394296 249765 > 5¹³² [i]