MinT - Best Known (169−45, 169, s)-Nets in Base 4

Best Known (169−45, 169, s)-Nets in Base 4

(169−45, 169, 531)-Net over F₄ — Constructive and digital

Digital (124, 169, 531)-net over F₄, using

t-expansion [i] based on digital (123, 169, 531)-net over F₄, using
- 5 times m-reduction [i] based on digital (123, 174, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 58, 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]

(169−45, 169, 576)-Net in Base 4 — Constructive

(124, 169, 576)-net in base 4, using

4¹ times duplication [i] based on (123, 168, 576)-net in base 4, using
- trace code for nets [i] based on (11, 56, 192)-net in base 64, using
  - base change [i] based on digital (3, 48, 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]

(169−45, 169, 1203)-Net over F₄ — Digital

Digital (124, 169, 1203)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(169−45, 169, 119458)-Net in Base 4 — Upper bound on s

There is no (124, 169, 119459)-net in base 4, because

1 times m-reduction [i] would yield (124, 168, 119459)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 140004 631542 867686 946943 795794 672470 622111 474232 674804 796954 105206 881702 115859 511896 207949 941921 637080 > 4¹⁶⁸ [i]