MinT - Best Known (243−99, 243, s)-Nets in Base 3

Best Known (243−99, 243, s)-Nets in Base 3

(243−99, 243, 156)-Net over F₃ — Constructive and digital

Digital (144, 243, 156)-net over F₃, using

1 times m-reduction [i] based on digital (144, 244, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 122, 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]

(243−99, 243, 211)-Net over F₃ — Digital

Digital (144, 243, 211)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(243−99, 243, 2123)-Net in Base 3 — Upper bound on s

There is no (144, 243, 2124)-net in base 3, because

1 times m-reduction [i] would yield (144, 242, 2124)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 29 718133 466318 696599 811257 486376 986427 322561 582769 010852 698498 151120 266145 083200 596327 656834 160246 103548 393179 829913 > 3²⁴² [i]