MinT - Best Known (66, 66+81, s)-Nets in Base 5

Best Known (66, 66+81, s)-Nets in Base 5

(66, 66+81, 82)-Net over F₅ — Constructive and digital

Digital (66, 147, 82)-net over F₅, using

t-expansion [i] based on digital (48, 147, 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]

(66, 66+81, 120)-Net over F₅ — Digital

Digital (66, 147, 120)-net over F₅, using

t-expansion [i] based on digital (61, 147, 120)-net over F₅, using
- net from sequence [i] based on digital (61, 119)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 61 and N(F) ≥ 120, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(66, 66+81, 1373)-Net in Base 5 — Upper bound on s

There is no (66, 147, 1374)-net in base 5, because

1 times m-reduction [i] would yield (66, 146, 1374)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1 139047 883537 408276 093405 816334 191156 465920 951761 229151 371588 776828 143736 698684 668737 898346 609813 106561 > 5¹⁴⁶ [i]