MinT - Best Known (243−147, 243, s)-Nets in Base 3

Best Known (243−147, 243, s)-Nets in Base 3

(243−147, 243, 64)-Net over F₃ — Constructive and digital

Digital (96, 243, 64)-net over F₃, using

t-expansion [i] based on digital (89, 243, 64)-net over F₃, using
- net from sequence [i] based on digital (89, 63)-sequence over F₃, using
  - base reduction for sequences [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(243−147, 243, 96)-Net over F₃ — Digital

Digital (96, 243, 96)-net over F₃, using

t-expansion [i] based on digital (89, 243, 96)-net over F₃, using
- net from sequence [i] based on digital (89, 95)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 89 and N(F) ≥ 96, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(243−147, 243, 465)-Net in Base 3 — Upper bound on s

There is no (96, 243, 466)-net in base 3, because

1 times m-reduction [i] would yield (96, 242, 466)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 29 930898 824827 507154 698181 811385 367044 811765 898624 385752 331855 709211 821565 300451 445946 609026 782964 049316 898958 232917 > 3²⁴² [i]