MinT - Best Known (230−166, 230, s)-Nets in Base 4

Best Known (230−166, 230, s)-Nets in Base 4

Digital (64, 230, 66)-net over F₄, using

t-expansion [i] based on digital (49, 230, 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 (64, 230, 99)-net over F₄, using

t-expansion [i] based on digital (61, 230, 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 (64, 230, 368)-net over F₄, because

2 times m-reduction [i] would yield digital (64, 228, 368)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²²⁸, 368, F₄, 164) (dual of [368, 140, 165]-code), but
  - residual code [i] would yield OA(4⁶⁴, 203, S₄, 41), but
    - the linear programming bound shows that MÂ â‰¥ 9237 828678 014661 888733 201438 074087 629177 850315 553208 766694 988543 356653 154675 278877 490717 655040 / 25 848321 162508 053807 967718 560003 483887 415061 992757 599441 > 4⁶⁴ [i]

There is no (64, 230, 428)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 393703 379457 899000 488810 695152 194741 895706 416790 709474 352050 217833 299465 405474 259306 438970 872609 280393 798183 507127 230677 400556 756378 430300 > 4²³⁰ [i]