MinT - Best Known (123, 123+103, s)-Nets in Base 3

Best Known (123, 123+103, s)-Nets in Base 3

(123, 123+103, 85)-Net over F₃ — Constructive and digital

Digital (123, 226, 85)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (27, 78, 37)-net over F₃, using
  - net from sequence [i] based on digital (27, 36)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 26, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]
2. digital (45, 148, 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]

(123, 123+103, 144)-Net over F₃ — Digital

Digital (123, 226, 144)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(123, 123+103, 1214)-Net in Base 3 — Upper bound on s

There is no (123, 226, 1215)-net in base 3, because

1 times m-reduction [i] would yield (123, 225, 1215)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 230592 662000 581927 370356 653625 360256 238992 668709 042377 806820 357899 870837 205875 906108 130990 261806 876524 099227 > 3²²⁵ [i]