MinT - Best Known (66, 66+61, s)-Nets in Base 5

Best Known (66, 66+61, s)-Nets in Base 5

(66, 66+61, 92)-Net over F₅ — Constructive and digital

Digital (66, 127, 92)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (5, 35, 20)-net over F₅, using
  - net from sequence [i] based on digital (5, 19)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 5 and N(F) ≥ 20, using
      - an explicitly constructive algebraic function field [i]
2. digital (31, 92, 72)-net over F₅, using
  - net from sequence [i] based on digital (31, 71)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 31 and N(F) ≥ 72, using
      - an explicitly constructive algebraic function field [i]

(66, 66+61, 145)-Net over F₅ — Digital

Digital (66, 127, 145)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(66, 66+61, 2574)-Net in Base 5 — Upper bound on s

There is no (66, 127, 2575)-net in base 5, because

1 times m-reduction [i] would yield (66, 126, 2575)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 11802 243757 397284 188549 639174 105397 028271 067089 508098 472812 092069 116739 188349 107844 342153 > 5¹²⁶ [i]