MinT - Best Known (137, 137+64, s)-Nets in Base 3

Best Known (137, 137+64, s)-Nets in Base 3

(137, 137+64, 164)-Net over F₃ — Constructive and digital

Digital (137, 201, 164)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (7, 39, 16)-net over F₃, using
  - net from sequence [i] based on digital (7, 15)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 7 and N(F) ≥ 16, using
      - an explicitly constructive algebraic function field [i]
2. digital (98, 162, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 81, 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]

(137, 137+64, 370)-Net over F₃ — Digital

Digital (137, 201, 370)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+64, 6318)-Net in Base 3 — Upper bound on s

There is no (137, 201, 6319)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 798912 257573 383324 784707 318488 456473 484082 438319 962575 400172 804900 116944 644822 994819 889102 439361 > 3²⁰¹ [i]