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

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

(68, 68+68, 88)-Net over F₅ — Constructive and digital

Digital (68, 136, 88)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (3, 37, 16)-net over F₅, using
  - net from sequence [i] based on digital (3, 15)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 3 and N(F) ≥ 16, using
      - an explicitly constructive algebraic function field [i]
2. digital (31, 99, 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+68, 134)-Net over F₅ — Digital

Digital (68, 136, 134)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+68, 2090)-Net in Base 5 — Upper bound on s

There is no (68, 136, 2091)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 116593 693630 737415 253542 415926 703213 531133 362779 003657 593980 440388 293850 507340 611202 531356 598265 > 5¹³⁶ [i]