MinT - Best Known (222−48, 222, s)-Nets in Base 3

Best Known (222−48, 222, s)-Nets in Base 3

(222−48, 222, 688)-Net over F₃ — Constructive and digital

Digital (174, 222, 688)-net over F₃, using

3² times duplication [i] based on digital (172, 220, 688)-net over F₃, using
- trace code for nets [i] based on digital (7, 55, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(222−48, 222, 1670)-Net over F₃ — Digital

Digital (174, 222, 1670)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(222−48, 222, 126947)-Net in Base 3 — Upper bound on s

There is no (174, 222, 126948)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8336 122386 343820 873213 148683 588640 993808 617894 800007 971860 829585 415752 675576 585820 331715 103757 324983 860289 > 3²²² [i]