MinT - Best Known (231−86, 231, s)-Nets in Base 3

Best Known (231−86, 231, s)-Nets in Base 3

(231−86, 231, 156)-Net over F₃ — Constructive and digital

Digital (145, 231, 156)-net over F₃, using

15 times m-reduction [i] based on digital (145, 246, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 123, 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]

(231−86, 231, 261)-Net over F₃ — Digital

Digital (145, 231, 261)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(231−86, 231, 3045)-Net in Base 3 — Upper bound on s

There is no (145, 231, 3046)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 166 274234 852968 363204 200980 553928 347227 757983 437538 222444 233540 915163 373248 153467 312130 502106 654130 616334 001745 > 3²³¹ [i]