MinT - Best Known (248−113, 248, s)-Nets in Base 3

Best Known (248−113, 248, s)-Nets in Base 3

(248−113, 248, 86)-Net over F₃ — Constructive and digital

Digital (135, 248, 86)-net over F₃, using

t-expansion [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]

(248−113, 248, 157)-Net over F₃ — Digital

Digital (135, 248, 157)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(248−113, 248, 1325)-Net in Base 3 — Upper bound on s

There is no (135, 248, 1326)-net in base 3, because

1 times m-reduction [i] would yield (135, 247, 1326)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7070 194829 017686 236071 054052 946420 558564 107731 824094 545428 346884 043198 915399 263889 833800 922924 827478 483003 829457 163313 > 3²⁴⁷ [i]