MinT - Best Known (245−111, 245, s)-Nets in Base 3

Best Known (245−111, 245, s)-Nets in Base 3

(245−111, 245, 86)-Net over F₃ — Constructive and digital

Digital (134, 245, 86)-net over F₃, using

3 times m-reduction [i] based on digital (134, 248, 86)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (32, 89, 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, 159, 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]

(245−111, 245, 158)-Net over F₃ — Digital

Digital (134, 245, 158)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(245−111, 245, 1342)-Net in Base 3 — Upper bound on s

There is no (134, 245, 1343)-net in base 3, because

1 times m-reduction [i] would yield (134, 244, 1343)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 270 691227 933377 264313 805789 029175 891158 944405 308965 488524 535347 833861 580682 742712 832341 030684 693428 648761 696612 223123 > 3²⁴⁴ [i]