MinT - Best Known (66, 66+77, s)-Nets in Base 5

Best Known (66, 66+77, s)-Nets in Base 5

(66, 66+77, 82)-Net over F₅ — Constructive and digital

Digital (66, 143, 82)-net over F₅, using

t-expansion [i] based on digital (48, 143, 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+77, 120)-Net over F₅ — Digital

Digital (66, 143, 120)-net over F₅, using

t-expansion [i] based on digital (61, 143, 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+77, 1509)-Net in Base 5 — Upper bound on s

There is no (66, 143, 1510)-net in base 5, because

1 times m-reduction [i] would yield (66, 142, 1510)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1828 795144 540777 651025 583121 574165 234302 569939 989371 560790 588637 895352 824528 018754 691433 178397 801025 > 5¹⁴² [i]