MinT - Best Known (55, 55+97, s)-Nets in Base 3

Best Known (55, 55+97, s)-Nets in Base 3

Digital (55, 152, 48)-net over F₃, using

Digital (55, 152, 64)-net over F₃, using

t-expansion [i] based on digital (49, 152, 64)-net over F₃, using
- net from sequence [i] based on digital (49, 63)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
    - a curve from the manYPoints database [i]

There is no digital (55, 152, 233)-net over F₃, because

1 times m-reduction [i] would yield digital (55, 151, 233)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁵¹, 233, F₃, 96) (dual of [233, 82, 97]-code), but
  - residual code [i] would yield OA(3⁵⁵, 136, S₃, 32), but
    - the linear programming bound shows that MÂ â‰¥ 3692 381881 435086 561083 538512 781551 192892 214805 966690 241846 076111 760291 378162 952668 981065 231048 784442 555038 600643 905023 157760 / 19 467037 909253 478265 941280 359023 466094 270607 906474 140765 003376 713759 364882 992681 224990 199775 364421 > 3⁵⁵ [i]

There is no (55, 152, 253)-net in base 3, because

1 times m-reduction [i] would yield (55, 151, 253)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 227760 419068 549113 267486 229448 111631 925963 283197 836539 127338 780915 636033 > 3¹⁵¹ [i]