MinT - Best Known (222−60, 222, s)-Nets in Base 4

Best Known (222−60, 222, s)-Nets in Base 4

(222−60, 222, 531)-Net over F₄ — Constructive and digital

Digital (162, 222, 531)-net over F₄, using

t-expansion [i] based on digital (161, 222, 531)-net over F₄, using
- 9 times m-reduction [i] based on digital (161, 231, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 77, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(222−60, 222, 576)-Net in Base 4 — Constructive

(162, 222, 576)-net in base 4, using

trace code for nets [i] based on (14, 74, 192)-net in base 64, using
- 3 times m-reduction [i] based on (14, 77, 192)-net in base 64, using
  - base change [i] based on digital (3, 66, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(222−60, 222, 1430)-Net over F₄ — Digital

Digital (162, 222, 1430)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(222−60, 222, 114503)-Net in Base 4 — Upper bound on s

There is no (162, 222, 114504)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 45 434370 915099 443669 322375 272310 478111 782178 662769 266168 182198 116373 771655 606528 246842 754818 212622 190500 208482 845774 280102 365794 936616 > 4²²² [i]