MinT - Best Known (242−154, 242, s)-Nets in Base 3

Best Known (242−154, 242, s)-Nets in Base 3

Digital (88, 242, 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]

Digital (88, 242, 84)-net over F₃, using

There is no digital (88, 242, 393)-net over F₃, because

1 times m-reduction [i] would yield digital (88, 241, 393)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²⁴¹, 393, F₃, 153) (dual of [393, 152, 154]-code), but
  - residual code [i] would yield linear OA(3⁸⁸, 239, F₃, 51) (dual of [239, 151, 52]-code), but
    - 1 times truncation [i] would yield linear OA(3⁸⁷, 238, F₃, 50) (dual of [238, 151, 51]-code), but
      - the Johnson bound shows that NÂ â‰¤ 1 109935 900942 577097 815120 032765 202024 203565 946556 215381 480293 664347 835582 < 3¹⁵¹ [i]

There is no (88, 242, 395)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 34 159662 169015 432373 904253 830441 535166 133693 077530 198494 858458 652627 570808 422329 594661 979305 639215 456246 615470 379007 > 3²⁴² [i]