MinT - Best Known (88, 88+158, s)-Nets in Base 3

Best Known (88, 88+158, s)-Nets in Base 3

(88, 88+158, 63)-Net over F₃ — Constructive and digital

Digital (88, 246, 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]

(88, 88+158, 84)-Net over F₃ — Digital

Digital (88, 246, 84)-net over F₃, using

t-expansion [i] based on digital (71, 246, 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]

(88, 88+158, 380)-Net over F₃ — Upper bound on s (digital)

There is no digital (88, 246, 381)-net over F₃, because

2 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]

(88, 88+158, 389)-Net in Base 3 — Upper bound on s

There is no (88, 246, 390)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2768 460503 245236 295971 671953 128418 543721 886284 408464 896751 548079 143997 442752 880010 270460 468819 638138 610083 894118 095177 > 3²⁴⁶ [i]