MinT - Best Known (172−83, 172, s)-Nets in Base 3

Best Known (172−83, 172, s)-Nets in Base 3

(172−83, 172, 69)-Net over F₃ — Constructive and digital

Digital (89, 172, 69)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (21, 62, 32)-net over F₃, using
  - net from sequence [i] based on digital (21, 31)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
      - an explicitly constructive algebraic function field [i]
2. digital (27, 110, 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]

(172−83, 172, 97)-Net over F₃ — Digital

Digital (89, 172, 97)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−83, 172, 748)-Net in Base 3 — Upper bound on s

There is no (89, 172, 749)-net in base 3, because

1 times m-reduction [i] would yield (89, 171, 749)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3892 163102 882772 425104 308600 535745 011612 543862 523334 967799 952888 073770 367836 307259 > 3¹⁷¹ [i]