MinT - Best Known (68, 68+49, s)-Nets in Base 5

Best Known (68, 68+49, s)-Nets in Base 5

(68, 68+49, 208)-Net over F₅ — Constructive and digital

Digital (68, 117, 208)-net over F₅, using

1 times m-reduction [i] based on digital (68, 118, 208)-net over F₅, using
- trace code for nets [i] based on digital (9, 59, 104)-net over F₂₅, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by GarcÃa and Quoos [i]

(68, 68+49, 220)-Net over F₅ — Digital

Digital (68, 117, 220)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+49, 5839)-Net in Base 5 — Upper bound on s

There is no (68, 117, 5840)-net in base 5, because

1 times m-reduction [i] would yield (68, 116, 5840)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1207 246138 892190 027230 306072 358692 108317 735420 652918 315291 092212 445516 658768 738305 > 5¹¹⁶ [i]