MinT - Best Known (67, 67+171, s)-Nets in Base 4

Best Known (67, 67+171, s)-Nets in Base 4

Digital (67, 238, 66)-net over F₄, using

t-expansion [i] based on digital (49, 238, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

Digital (67, 238, 99)-net over F₄, using

t-expansion [i] based on digital (61, 238, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

There is no digital (67, 238, 399)-net over F₄, because

3 times m-reduction [i] would yield digital (67, 235, 399)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²³⁵, 399, F₄, 168) (dual of [399, 164, 169]-code), but
  - residual code [i] would yield OA(4⁶⁷, 230, S₄, 42), but
    - the linear programming bound shows that MÂ â‰¥ 133 864877 868223 848277 340960 330245 263082 378026 507294 938785 509330 612908 753764 170596 352000 / 6063 223087 091274 607869 862566 805531 022255 644183 > 4⁶⁷ [i]

There is no (67, 238, 450)-net in base 4, because

1 times m-reduction [i] would yield (67, 237, 450)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 55818 026006 333072 784981 358460 843727 997532 259199 992801 468598 665981 066389 760978 448265 708701 621088 262048 308497 711258 240398 369511 486559 731006 199200 > 4²³⁷ [i]