MinT - Best Known (68, 68+61, s)-Nets in Base 5

Best Known (68, 68+61, s)-Nets in Base 5

(68, 68+61, 104)-Net over F₅ — Constructive and digital

Digital (68, 129, 104)-net over F₅, using

1 times m-reduction [i] based on digital (68, 130, 104)-net over F₅, using
- trace code for nets [i] based on digital (3, 65, 52)-net over F₂₅, using
  - net from sequence [i] based on digital (3, 51)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 3 and N(F) ≥ 52, using
      - a function field by GarcÃa and Quoos [i]

(68, 68+61, 155)-Net over F₅ — Digital

Digital (68, 129, 155)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+61, 2868)-Net in Base 5 — Upper bound on s

There is no (68, 129, 2869)-net in base 5, because

1 times m-reduction [i] would yield (68, 128, 2869)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 294654 531531 226734 222734 204035 196430 668624 807451 768595 020822 095591 992979 198533 243543 511337 > 5¹²⁸ [i]