MinT - Best Known (67, 67+177, s)-Nets in Base 4

Best Known (67, 67+177, s)-Nets in Base 4

Digital (67, 244, 66)-net over F₄, using

t-expansion [i] based on digital (49, 244, 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, 244, 99)-net over F₄, using

t-expansion [i] based on digital (61, 244, 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, 244, 362)-net over F₄, because

1 times m-reduction [i] would yield digital (67, 243, 362)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁴³, 362, F₄, 176) (dual of [362, 119, 177]-code), but
  - residual code [i] would yield OA(4⁶⁷, 185, S₄, 44), but
    - the linear programming bound shows that MÂ â‰¥ 27 922736 614479 963563 498030 039555 280180 497392 114764 937434 686895 163379 799248 329552 391127 539087 989670 936576 / 1212 198916 755108 948361 633672 026124 789528 916769 954260 490443 945723 > 4⁶⁷ [i]

There is no (67, 244, 446)-net in base 4, because

1 times m-reduction [i] would yield (67, 243, 446)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 234 616739 815577 135331 379144 171411 953764 086479 606398 350919 175399 850276 387908 007436 296498 747223 409815 564517 688698 291532 758258 077916 279174 534813 257950 > 4²⁴³ [i]