MinT - Best Known (68, 68+65, s)-Nets in Base 5

Best Known (68, 68+65, s)-Nets in Base 5

(68, 68+65, 92)-Net over F₅ — Constructive and digital

Digital (68, 133, 92)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (5, 37, 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, 96, 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]

(68, 68+65, 142)-Net over F₅ — Digital

Digital (68, 133, 142)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+65, 2420)-Net in Base 5 — Upper bound on s

There is no (68, 133, 2421)-net in base 5, because

1 times m-reduction [i] would yield (68, 132, 2421)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 185 085271 240043 967954 388126 149768 920007 009453 087392 618302 427874 431097 812962 231697 293149 903745 > 5¹³² [i]