MinT - Best Known (193−67, 193, s)-Nets in Base 3

Best Known (193−67, 193, s)-Nets in Base 3

(193−67, 193, 156)-Net over F₃ — Constructive and digital

Digital (126, 193, 156)-net over F₃, using

15 times m-reduction [i] based on digital (126, 208, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 104, 78)-net over F₉, using
  - net from sequence [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(193−67, 193, 273)-Net over F₃ — Digital

Digital (126, 193, 273)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(193−67, 193, 3896)-Net in Base 3 — Upper bound on s

There is no (126, 193, 3897)-net in base 3, because

1 times m-reduction [i] would yield (126, 192, 3897)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 40 553385 318775 916506 147095 255577 413722 547174 545662 690732 049810 305140 785253 584423 487893 364723 > 3¹⁹² [i]