MinT - Best Known (249−111, 249, s)-Nets in Base 3

Best Known (249−111, 249, s)-Nets in Base 3

(249−111, 249, 128)-Net over F₃ — Constructive and digital

Digital (138, 249, 128)-net over F₃, using

1 times m-reduction [i] based on digital (138, 250, 128)-net over F₃, using
- trace code for nets [i] based on digital (13, 125, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(249−111, 249, 167)-Net over F₃ — Digital

Digital (138, 249, 167)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(249−111, 249, 1458)-Net in Base 3 — Upper bound on s

There is no (138, 249, 1459)-net in base 3, because

1 times m-reduction [i] would yield (138, 248, 1459)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 21843 725920 439370 590252 312435 201602 645466 667189 545817 005690 331999 579740 285474 316378 003733 932936 019751 949320 339893 080675 > 3²⁴⁸ [i]