MinT - Best Known (245−157, 245, s)-Nets in Base 3

Best Known (245−157, 245, s)-Nets in Base 3

(245−157, 245, 63)-Net over F₃ — Constructive and digital

Digital (88, 245, 63)-net over F₃, using

net from sequence [i] based on digital (88, 62)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 62)-sequence over F₉, using
  - s-reduction based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(245−157, 245, 84)-Net over F₃ — Digital

Digital (88, 245, 84)-net over F₃, using

t-expansion [i] based on digital (71, 245, 84)-net over F₃, using
- net from sequence [i] based on digital (71, 83)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(245−157, 245, 380)-Net over F₃ — Upper bound on s (digital)

There is no digital (88, 245, 381)-net over F₃, because

1 times m-reduction [i] would yield digital (88, 244, 381)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²⁴⁴, 381, F₃, 156) (dual of [381, 137, 157]-code), but
  - residual code [i] would yield linear OA(3⁸⁸, 224, F₃, 52) (dual of [224, 136, 53]-code), but
    - the Johnson bound shows that NÂ â‰¤ 73138 750163 909178 998184 764057 381403 209399 446777 305834 354572 535302 < 3¹³⁶ [i]

(245−157, 245, 391)-Net in Base 3 — Upper bound on s

There is no (88, 245, 392)-net in base 3, because

1 times m-reduction [i] would yield (88, 244, 392)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 285 539408 636894 817729 503166 297106 989773 803226 951402 187488 869004 523821 189737 817309 948824 186679 643033 119864 167037 272497 > 3²⁴⁴ [i]