MinT - Best Known (162−41, 162, s)-Nets in Base 4

Best Known (162−41, 162, s)-Nets in Base 4

(162−41, 162, 531)-Net over F₄ — Constructive and digital

Digital (121, 162, 531)-net over F₄, using

9 times m-reduction [i] based on digital (121, 171, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 57, 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]

(162−41, 162, 648)-Net in Base 4 — Constructive

(121, 162, 648)-net in base 4, using

trace code for nets [i] based on (13, 54, 216)-net in base 64, using
- 2 times m-reduction [i] based on (13, 56, 216)-net in base 64, using
  - base change [i] based on digital (5, 48, 216)-net over F₁₂₈, using
    - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(162−41, 162, 1462)-Net over F₄ — Digital

Digital (121, 162, 1462)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(162−41, 162, 194415)-Net in Base 4 — Upper bound on s

There is no (121, 162, 194416)-net in base 4, because

1 times m-reduction [i] would yield (121, 161, 194416)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8 544030 106965 903014 836481 219279 301127 632544 143818 732410 457050 710240 628116 482918 776658 922978 156934 > 4¹⁶¹ [i]