MinT - Best Known (230−85, 230, s)-Nets in Base 3

Best Known (230−85, 230, s)-Nets in Base 3

(230−85, 230, 156)-Net over F₃ — Constructive and digital

Digital (145, 230, 156)-net over F₃, using

16 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]

(230−85, 230, 265)-Net over F₃ — Digital

Digital (145, 230, 265)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(230−85, 230, 3256)-Net in Base 3 — Upper bound on s

There is no (145, 230, 3257)-net in base 3, because

1 times m-reduction [i] would yield (145, 229, 3257)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 18 351725 270143 354654 525546 123594 590090 954550 424095 550841 258774 861299 512911 437266 230387 596295 215563 857578 712153 > 3²²⁹ [i]