MinT - Best Known (65, 65+61, s)-Nets in Base 5

Best Known (65, 65+61, s)-Nets in Base 5

(65, 65+61, 90)-Net over F₅ — Constructive and digital

Digital (65, 126, 90)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (4, 34, 18)-net over F₅, using
  - net from sequence [i] based on digital (4, 17)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 4 and N(F) ≥ 18, 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]

(65, 65+61, 141)-Net over F₅ — Digital

Digital (65, 126, 141)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(65, 65+61, 2439)-Net in Base 5 — Upper bound on s

There is no (65, 126, 2440)-net in base 5, because

1 times m-reduction [i] would yield (65, 125, 2440)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 2379 085486 040205 389689 684993 560548 223900 840456 615807 055075 185206 729649 208834 370953 615489 > 5¹²⁵ [i]