MinT - Best Known (101−65, 101, s)-Nets in Base 4

Best Known (101−65, 101, s)-Nets in Base 4

Digital (36, 101, 56)-net over F₄, using

t-expansion [i] based on digital (33, 101, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
    - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

Digital (36, 101, 65)-net over F₄, using

t-expansion [i] based on digital (33, 101, 65)-net over F₄, using
- net from sequence [i] based on digital (33, 64)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 65, using
    - Drinfeld modules of rank 1 [i]

There is no (36, 101, 299)-net in base 4, because

1 times m-reduction [i] would yield (36, 100, 299)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹⁰⁰, 299, S₄, 64), but
  - 1 times code embedding in larger space [i] would yield OA(4¹⁰¹, 300, S₄, 64), but
    - the linear programming bound shows that MÂ â‰¥ 1 231265 431537 788363 998593 064935 723340 754119 840201 132340 137144 126545 911415 806525 100699 435502 000219 063131 921131 435910 945455 478688 417027 331901 674121 045545 301803 617922 641716 838400 / 170245 555311 187291 018741 043496 578809 211994 161262 146673 891958 873123 481881 751387 033606 486355 185045 431953 178363 079839 > 4¹⁰¹ [i]