MinT - Best Known (135−71, 135, s)-Nets in Base 5

Best Known (135−71, 135, s)-Nets in Base 5

(135−71, 135, 82)-Net over F₅ — Constructive and digital

Digital (64, 135, 82)-net over F₅, using

t-expansion [i] based on digital (48, 135, 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]

(135−71, 135, 120)-Net over F₅ — Digital

Digital (64, 135, 120)-net over F₅, using

t-expansion [i] based on digital (61, 135, 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]

(135−71, 135, 1623)-Net in Base 5 — Upper bound on s

There is no (64, 135, 1624)-net in base 5, because

1 times m-reduction [i] would yield (64, 134, 1624)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 4636 482654 676341 047858 984963 835707 891342 976427 294977 757176 322519 762881 223930 555616 527037 082721 > 5¹³⁴ [i]