MinT - Best Known (57, 207, s)-Nets in Base 4

Best Known (57, 207, s)-Nets in Base 4

Digital (57, 207, 66)-net over F₄, using

t-expansion [i] based on digital (49, 207, 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 (57, 207, 91)-net over F₄, using

t-expansion [i] based on digital (50, 207, 91)-net over F₄, using
- net from sequence [i] based on digital (50, 90)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 50 and N(F) ≥ 91, using
    - a curve from the manYPoints database [i]

There is no digital (57, 207, 326)-net over F₄, because

2 times m-reduction [i] would yield digital (57, 205, 326)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁰⁵, 326, F₄, 148) (dual of [326, 121, 149]-code), but
  - residual code [i] would yield OA(4⁵⁷, 177, S₄, 37), but
    - the linear programming bound shows that MÂ â‰¥ 7 916017 728459 289141 982805 636437 938856 538720 259539 351050 169416 698004 574623 120920 412160 / 363 505843 153703 749393 199977 171265 253621 464016 748729 > 4⁵⁷ [i]

There is no (57, 207, 381)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 43692 411990 686562 954214 934273 060902 937668 658533 332212 465306 579567 134135 140051 966249 482931 820155 008911 126541 518828 089360 317404 > 4²⁰⁷ [i]