MinT - Best Known (162, 162+59, s)-Nets in Base 4

Best Known (162, 162+59, s)-Nets in Base 4

(162, 162+59, 531)-Net over F₄ — Constructive and digital

Digital (162, 221, 531)-net over F₄, using

t-expansion [i] based on digital (161, 221, 531)-net over F₄, using
- 10 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]

(162, 162+59, 576)-Net in Base 4 — Constructive

(162, 221, 576)-net in base 4, using

1 times m-reduction [i] based on (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]

(162, 162+59, 1502)-Net over F₄ — Digital

Digital (162, 221, 1502)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(162, 162+59, 143641)-Net in Base 4 — Upper bound on s

There is no (162, 221, 143642)-net in base 4, because

1 times m-reduction [i] would yield (162, 220, 143642)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 839319 189913 217320 576798 909814 610194 914222 940947 414165 680242 358559 607359 742788 824443 327050 111857 591278 405327 071477 639645 806682 456256 > 4²²⁰ [i]