MinT - Best Known (162−43, 162, s)-Nets in Base 4

Best Known (162−43, 162, s)-Nets in Base 4

(162−43, 162, 531)-Net over F₄ — Constructive and digital

Digital (119, 162, 531)-net over F₄, using

6 times m-reduction [i] based on digital (119, 168, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 56, 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]

(162−43, 162, 576)-Net in Base 4 — Constructive

(119, 162, 576)-net in base 4, using

trace code for nets [i] based on (11, 54, 192)-net in base 64, using
- 2 times m-reduction [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]

(162−43, 162, 1177)-Net over F₄ — Digital

Digital (119, 162, 1177)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(162−43, 162, 119427)-Net in Base 4 — Upper bound on s

There is no (119, 162, 119428)-net in base 4, because

1 times m-reduction [i] would yield (119, 161, 119428)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8 545349 632307 118224 783389 184125 764751 677816 776629 362194 832686 032660 690267 315841 057967 495329 863840 > 4¹⁶¹ [i]