MinT - Best Known (63, 234, s)-Nets in Base 4

Best Known (63, 234, s)-Nets in Base 4

Digital (63, 234, 66)-net over F₄, using

t-expansion [i] based on digital (49, 234, 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 (63, 234, 99)-net over F₄, using

t-expansion [i] based on digital (61, 234, 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 (63, 234, 334)-net over F₄, because

3 times m-reduction [i] would yield digital (63, 231, 334)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²³¹, 334, F₄, 168) (dual of [334, 103, 169]-code), but
  - residual code [i] would yield OA(4⁶³, 165, S₄, 42), but
    - the linear programming bound shows that MÂ â‰¥ 34 077190 942729 726882 024839 335902 035522 717143 016283 935337 510998 524998 580004 361097 586089 334532 997120 / 383476 318008 814835 283839 224897 225684 554347 388608 827379 925139 > 4⁶³ [i]

There is no (63, 234, 417)-net in base 4, because

1 times m-reduction [i] would yield (63, 233, 417)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 197 195029 786515 525692 972211 707505 145477 180366 232362 842248 148657 429280 957103 442643 154864 668163 992460 195681 134700 869936 103241 161687 166791 532160 > 4²³³ [i]