MinT - Best Known (215−71, 215, s)-Nets in Base 3

Best Known (215−71, 215, s)-Nets in Base 3

(215−71, 215, 162)-Net over F₃ — Constructive and digital

Digital (144, 215, 162)-net over F₃, using

9 times m-reduction [i] based on digital (144, 224, 162)-net over F₃, using
- trace code for nets [i] based on digital (32, 112, 81)-net over F₉, using
  - net from sequence [i] based on digital (32, 80)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
      - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(215−71, 215, 349)-Net over F₃ — Digital

Digital (144, 215, 349)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(215−71, 215, 5713)-Net in Base 3 — Upper bound on s

There is no (144, 215, 5714)-net in base 3, because

1 times m-reduction [i] would yield (144, 214, 5714)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 276432 924192 315163 934209 416821 232929 409805 937961 244807 030933 501776 149308 904053 812860 232042 955240 069073 > 3²¹⁴ [i]