MinT - Best Known (133, 133+111, s)-Nets in Base 3

Best Known (133, 133+111, s)-Nets in Base 3

(133, 133+111, 86)-Net over F₃ — Constructive and digital

Digital (133, 244, 86)-net over F₃, using

1 times m-reduction [i] based on digital (133, 245, 86)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (32, 88, 38)-net over F₃, using
    - net from sequence [i] based on digital (32, 37)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, using
        an explicitly constructive algebraic function field [i]
  2. digital (45, 157, 48)-net over F₃, using
    - net from sequence [i] based on digital (45, 47)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
        an explicitly constructive algebraic function field [i]

(133, 133+111, 155)-Net over F₃ — Digital

Digital (133, 244, 155)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(133, 133+111, 1314)-Net in Base 3 — Upper bound on s

There is no (133, 244, 1315)-net in base 3, because

1 times m-reduction [i] would yield (133, 243, 1315)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 88 803337 268122 539101 840083 342867 624907 459709 201262 420199 058888 576224 555066 738877 354356 610700 769151 306388 848317 365539 > 3²⁴³ [i]