MinT - Best Known (44, 44+61, s)-Nets in Base 5

Best Known (44, 44+61, s)-Nets in Base 5

(44, 44+61, 78)-Net over F₅ — Constructive and digital

Digital (44, 105, 78)-net over F₅, using

t-expansion [i] based on digital (38, 105, 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]

(44, 44+61, 84)-Net over F₅ — Digital

Digital (44, 105, 84)-net over F₅, using

t-expansion [i] based on digital (43, 105, 84)-net over F₅, using
- net from sequence [i] based on digital (43, 83)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 43 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(44, 44+61, 775)-Net in Base 5 — Upper bound on s

There is no (44, 105, 776)-net in base 5, because

1 times m-reduction [i] would yield (44, 104, 776)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 4 951365 333895 980082 084016 003188 664354 791576 495745 315785 055435 985343 697025 > 5¹⁰⁴ [i]