MinT - Best Known (68, 68+57, s)-Nets in Base 5

Best Known (68, 68+57, s)-Nets in Base 5

(68, 68+57, 132)-Net over F₅ — Constructive and digital

Digital (68, 125, 132)-net over F₅, using

3 times m-reduction [i] based on digital (68, 128, 132)-net over F₅, using
- trace code for nets [i] based on digital (4, 64, 66)-net over F₂₅, using
  - net from sequence [i] based on digital (4, 65)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 4 and N(F) ≥ 66, using
      - a function field by GarcÃa and Quoos [i]

(68, 68+57, 171)-Net over F₅ — Digital

Digital (68, 125, 171)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+57, 3498)-Net in Base 5 — Upper bound on s

There is no (68, 125, 3499)-net in base 5, because

1 times m-reduction [i] would yield (68, 124, 3499)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 472 997138 670135 715675 966564 126739 838977 040523 416769 037974 381182 229974 036172 878870 086929 > 5¹²⁴ [i]