MinT - Best Known (160, 160+79, s)-Nets in Base 3

Best Known (160, 160+79, s)-Nets in Base 3

(160, 160+79, 164)-Net over F₃ — Constructive and digital

Digital (160, 239, 164)-net over F₃, using

3¹ times duplication [i] based on digital (159, 238, 164)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 46, 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 (113, 192, 148)-net over F₃, using
    - trace code for nets [i] based on digital (17, 96, 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]

(160, 160+79, 381)-Net over F₃ — Digital

Digital (160, 239, 381)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(160, 160+79, 6243)-Net in Base 3 — Upper bound on s

There is no (160, 239, 6244)-net in base 3, because

1 times m-reduction [i] would yield (160, 238, 6244)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 360983 546804 618994 889436 069020 839275 040830 399423 621035 577001 741641 415066 540202 488550 620451 174286 036895 641585 137521 > 3²³⁸ [i]