MinT - Best Known (232−166, 232, s)-Nets in Base 4

Best Known (232−166, 232, s)-Nets in Base 4

Digital (66, 232, 66)-net over F₄, using

t-expansion [i] based on digital (49, 232, 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 (66, 232, 99)-net over F₄, using

t-expansion [i] based on digital (61, 232, 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 (66, 232, 404)-net over F₄, because

2 times m-reduction [i] would yield digital (66, 230, 404)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²³⁰, 404, F₄, 164) (dual of [404, 174, 165]-code), but
  - residual code [i] would yield OA(4⁶⁶, 239, S₄, 41), but
    - the linear programming bound shows that MÂ â‰¥ 222 701100 724979 298348 829615 805692 012896 998205 225198 824371 851659 244379 751219 018386 636800 / 38358 477730 413015 488846 745362 992009 108847 310917 > 4⁶⁶ [i]

There is no (66, 232, 444)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 48 806534 982884 540465 585048 406117 382779 343761 559073 626168 844401 364757 202017 145485 853958 276606 660322 800217 434696 988774 308364 255384 437637 093370 > 4²³² [i]