MinT - Best Known (243−80, 243, s)-Nets in Base 3

Best Known (243−80, 243, s)-Nets in Base 3

(243−80, 243, 167)-Net over F₃ — Constructive and digital

Digital (163, 243, 167)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (9, 49, 19)-net over F₃, using
  - net from sequence [i] based on digital (9, 18)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 9 and N(F) ≥ 19, using
      - an explicitly constructive algebraic function field [i]
2. digital (114, 194, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 97, 74)-net over F₉, using
    - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
        a function field by GarcÃa and Quoos [i]

(243−80, 243, 392)-Net over F₃ — Digital

Digital (163, 243, 392)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(243−80, 243, 6201)-Net in Base 3 — Upper bound on s

There is no (163, 243, 6202)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 87 261810 306675 370342 722040 104192 628295 113104 114921 248085 092359 064112 992438 514317 251495 706934 588085 464886 407324 027153 > 3²⁴³ [i]