MinT - Best Known (131, 131+87, s)-Nets in Base 3

Best Known (131, 131+87, s)-Nets in Base 3

(131, 131+87, 156)-Net over F₃ — Constructive and digital

Digital (131, 218, 156)-net over F₃, using

trace code for nets [i] based on digital (22, 109, 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]

(131, 131+87, 203)-Net over F₃ — Digital

Digital (131, 218, 203)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(131, 131+87, 2116)-Net in Base 3 — Upper bound on s

There is no (131, 218, 2117)-net in base 3, because

1 times m-reduction [i] would yield (131, 217, 2117)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 34 462254 459785 609565 329216 597346 370366 053304 186839 021766 466733 203230 901445 311089 264628 669540 941616 404875 > 3²¹⁷ [i]