MinT - Best Known (254−182, 254, s)-Nets in Base 4

Best Known (254−182, 254, s)-Nets in Base 4

Digital (72, 254, 66)-net over F₄, using

t-expansion [i] based on digital (49, 254, 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 (72, 254, 105)-net over F₄, using

t-expansion [i] based on digital (70, 254, 105)-net over F₄, using
- net from sequence [i] based on digital (70, 104)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 70 and N(F) ≥ 105, using
    - Drinfeld modules of rank 1 [i]

There is no digital (72, 254, 431)-net over F₄, because

2 times m-reduction [i] would yield digital (72, 252, 431)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁵², 431, F₄, 180) (dual of [431, 179, 181]-code), but
  - residual code [i] would yield OA(4⁷², 250, S₄, 45), but
    - the linear programming bound shows that MÂ â‰¥ 844 253129 128337 968002 738138 104958 795392 851484 110411 486022 086287 533951 460726 969110 480212 292670 586880 / 36 177347 187970 342236 372937 633185 534214 836452 245739 305493 > 4⁷² [i]

There is no (72, 254, 482)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 839 123640 493358 490432 604711 464308 408313 531657 231526 082673 340664 540223 561763 536001 564391 044758 380718 395245 767051 082106 644490 930761 969444 550461 750529 293456 > 4²⁵⁴ [i]