MinT - Best Known (239−174, 239, s)-Nets in Base 4

Best Known (239−174, 239, s)-Nets in Base 4

Digital (65, 239, 66)-net over F₄, using

t-expansion [i] based on digital (49, 239, 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 (65, 239, 99)-net over F₄, using

t-expansion [i] based on digital (61, 239, 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 (65, 239, 347)-net over F₄, because

2 times m-reduction [i] would yield digital (65, 237, 347)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²³⁷, 347, F₄, 172) (dual of [347, 110, 173]-code), but
  - residual code [i] would yield OA(4⁶⁵, 174, S₄, 43), but
    - the linear programming bound shows that MÂ â‰¥ 2 356982 708196 944001 632405 639197 943867 825680 395816 503599 985988 433957 852565 175436 909425 167874 773494 005760 / 1698 395268 555289 237515 503113 977957 762702 705839 816361 669722 314819 > 4⁶⁵ [i]

There is no (65, 239, 431)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 879751 580800 826973 041734 519348 527562 097299 843525 761384 773938 825936 843652 879015 629161 826973 771923 321897 758181 387591 654828 200204 154758 459840 903280 > 4²³⁹ [i]