MinT - Best Known (66, 245, s)-Nets in Base 4

Best Known (66, 245, s)-Nets in Base 4

Digital (66, 245, 66)-net over F₄, using

t-expansion [i] based on digital (49, 245, 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, 245, 99)-net over F₄, using

t-expansion [i] based on digital (61, 245, 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, 245, 347)-net over F₄, because

3 times m-reduction [i] would yield digital (66, 242, 347)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁴², 347, F₄, 176) (dual of [347, 105, 177]-code), but
  - residual code [i] would yield OA(4⁶⁶, 170, S₄, 44), but
    - the linear programming bound shows that MÂ â‰¥ 1976 506064 459033 075362 451713 190875 400705 748869 321511 254504 692102 419708 690447 989326 466803 280840 783010 320768 871264 419840 000000 / 348910 981362 978418 180472 516419 977086 112239 745527 698751 797516 935464 412701 432423 842977 > 4⁶⁶ [i]

There is no (66, 245, 436)-net in base 4, because

1 times m-reduction [i] would yield (66, 244, 436)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 803 509214 801784 279291 829848 195619 650413 522369 283724 603604 794541 427056 960657 389316 565193 602291 785007 874340 977846 120857 680329 831511 293975 967241 127000 > 4²⁴⁴ [i]