MinT - Best Known (66, 66+176, s)-Nets in Base 4

Best Known (66, 66+176, s)-Nets in Base 4

Digital (66, 242, 66)-net over F₄, using

t-expansion [i] based on digital (49, 242, 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, 242, 99)-net over F₄, using

t-expansion [i] based on digital (61, 242, 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, 242, 347)-net over F₄, because

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, 242, 438)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 58 632488 190859 510302 230178 188209 206587 440049 449035 463996 602088 560772 713900 781841 508719 318600 098969 401551 298324 434643 834169 337204 583581 098732 037800 > 4²⁴² [i]