MinT - Best Known (68, 68+67, s)-Nets in Base 5

Best Known (68, 68+67, s)-Nets in Base 5

(68, 68+67, 90)-Net over F₅ — Constructive and digital

Digital (68, 135, 90)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (4, 37, 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, 98, 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+67, 136)-Net over F₅ — Digital

Digital (68, 135, 136)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+67, 2243)-Net in Base 5 — Upper bound on s

There is no (68, 135, 2244)-net in base 5, because

1 times m-reduction [i] would yield (68, 134, 2244)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 4639 931247 454548 580972 295229 111827 637604 729569 246502 772144 425583 887387 797784 025651 384047 552273 > 5¹³⁴ [i]