MinT - Best Known (209−149, 209, s)-Nets in Base 4

Best Known (209−149, 209, s)-Nets in Base 4

Digital (60, 209, 66)-net over F₄, using

t-expansion [i] based on digital (49, 209, 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 (60, 209, 91)-net over F₄, using

t-expansion [i] based on digital (50, 209, 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 (60, 209, 377)-net over F₄, because

1 times m-reduction [i] would yield digital (60, 208, 377)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁰⁸, 377, F₄, 148) (dual of [377, 169, 149]-code), but
  - residual code [i] would yield OA(4⁶⁰, 228, S₄, 37), but
    - the linear programming bound shows that MÂ â‰¥ 4 063856 865702 461109 434988 542147 550126 468452 401248 290460 178687 206981 384116 633600 / 3 020261 554987 735388 398172 544670 898492 334641 > 4⁶⁰ [i]

There is no (60, 209, 408)-net in base 4, because

1 times m-reduction [i] would yield (60, 208, 408)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 185619 007282 226833 756180 983692 523062 114289 410318 373113 629670 936584 915007 684857 710050 396934 793701 126638 341964 610541 366438 480420 > 4²⁰⁸ [i]