MinT - Best Known (218−70, 218, s)-Nets in Base 3

Best Known (218−70, 218, s)-Nets in Base 3

(218−70, 218, 167)-Net over F₃ — Constructive and digital

Digital (148, 218, 167)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (9, 44, 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 (104, 174, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 87, 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]

(218−70, 218, 386)-Net over F₃ — Digital

Digital (148, 218, 386)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(218−70, 218, 6482)-Net in Base 3 — Upper bound on s

There is no (148, 218, 6483)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 103 389441 887108 080104 556006 481696 371410 905579 295225 985439 339051 692570 379606 329460 187640 400383 802988 921771 > 3²¹⁸ [i]