MinT - Best Known (50, 50+63, s)-Nets in Base 5

Best Known (50, 50+63, s)-Nets in Base 5

(50, 50+63, 82)-Net over F₅ — Constructive and digital

Digital (50, 113, 82)-net over F₅, using

t-expansion [i] based on digital (48, 113, 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]

(50, 50+63, 96)-Net over F₅ — Digital

Digital (50, 113, 96)-net over F₅, using

t-expansion [i] based on digital (49, 113, 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]

(50, 50+63, 1018)-Net in Base 5 — Upper bound on s

There is no (50, 113, 1019)-net in base 5, because

1 times m-reduction [i] would yield (50, 112, 1019)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1 982585 171471 399747 666267 168336 703158 359481 945819 839514 291004 848147 300629 232565 > 5¹¹² [i]