MinT - Best Known (228−165, 228, s)-Nets in Base 4

Best Known (228−165, 228, s)-Nets in Base 4

Digital (63, 228, 66)-net over F₄, using

t-expansion [i] based on digital (49, 228, 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, 228, 99)-net over F₄, using

t-expansion [i] based on digital (61, 228, 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, 228, 350)-net over F₄, because

1 times m-reduction [i] would yield digital (63, 227, 350)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²²⁷, 350, F₄, 164) (dual of [350, 123, 165]-code), but
  - residual code [i] would yield OA(4⁶³, 185, S₄, 41), but
    - the linear programming bound shows that MÂ â‰¥ 61 899957 150953 106035 135074 637879 046641 791483 715273 894502 688876 097802 046438 481549 501189 324800 / 720746 857989 042001 933213 860469 455977 050620 069897 655507 > 4⁶³ [i]

There is no (63, 228, 421)-net in base 4, because

1 times m-reduction [i] would yield (63, 227, 421)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 51477 605557 233745 917127 999840 379429 878287 395774 624509 173349 363587 639788 030231 113597 002457 622257 983099 504023 777624 524775 542490 733848 666728 > 4²²⁷ [i]