MinT - Best Known (249−115, 249, s)-Nets in Base 3

Best Known (249−115, 249, s)-Nets in Base 3

(249−115, 249, 86)-Net over F₃ — Constructive and digital

Digital (134, 249, 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, 160, 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]

(249−115, 249, 152)-Net over F₃ — Digital

Digital (134, 249, 152)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(249−115, 249, 1259)-Net in Base 3 — Upper bound on s

There is no (134, 249, 1260)-net in base 3, because

1 times m-reduction [i] would yield (134, 248, 1260)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 21845 494487 592424 311485 388953 626855 970421 486894 315315 597938 838241 498679 846505 036348 530460 019361 923501 254646 110271 711065 > 3²⁴⁸ [i]