MinT - Best Known (50, 50+81, s)-Nets in Base 5

Best Known (50, 50+81, s)-Nets in Base 5

(50, 50+81, 82)-Net over F₅ — Constructive and digital

Digital (50, 131, 82)-net over F₅, using

t-expansion [i] based on digital (48, 131, 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+81, 96)-Net over F₅ — Digital

Digital (50, 131, 96)-net over F₅, using

t-expansion [i] based on digital (49, 131, 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+81, 707)-Net in Base 5 — Upper bound on s

There is no (50, 131, 708)-net in base 5, because

1 times m-reduction [i] would yield (50, 130, 708)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 7 438885 671088 577316 717367 320803 383612 934444 856789 217433 082233 767433 340163 818063 822523 218945 > 5¹³⁰ [i]