MinT - Best Known (223−59, 223, s)-Nets in Base 4

Best Known (223−59, 223, s)-Nets in Base 4

(223−59, 223, 531)-Net over F₄ — Constructive and digital

Digital (164, 223, 531)-net over F₄, using

t-expansion [i] based on digital (163, 223, 531)-net over F₄, using
- 11 times m-reduction [i] based on digital (163, 234, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 78, 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]

(223−59, 223, 576)-Net in Base 4 — Constructive

(164, 223, 576)-net in base 4, using

2 times m-reduction [i] based on (164, 225, 576)-net in base 4, using
- trace code for nets [i] based on (14, 75, 192)-net in base 64, using
  - 2 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]

(223−59, 223, 1574)-Net over F₄ — Digital

Digital (164, 223, 1574)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(223−59, 223, 158055)-Net in Base 4 — Upper bound on s

There is no (164, 223, 158056)-net in base 4, because

1 times m-reduction [i] would yield (164, 222, 158056)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 45 434085 483268 266826 543547 876548 615451 930504 037904 150370 883758 672703 854432 693521 105383 568089 946526 490243 637485 791125 235568 998331 420340 > 4²²² [i]