MinT - Best Known (67, 67+64, s)-Nets in Base 5

Best Known (67, 67+64, s)-Nets in Base 5

(67, 67+64, 90)-Net over F₅ — Constructive and digital

Digital (67, 131, 90)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (4, 36, 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, 95, 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]

(67, 67+64, 141)-Net over F₅ — Digital

Digital (67, 131, 141)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(67, 67+64, 2300)-Net in Base 5 — Upper bound on s

There is no (67, 131, 2301)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 36 964753 715320 070012 227931 334946 573312 133797 148973 140402 690612 458398 471931 591870 409884 337025 > 5¹³¹ [i]