MinT - Best Known (239−170, 239, s)-Nets in Base 4

Best Known (239−170, 239, s)-Nets in Base 4

Digital (69, 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 (69, 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 (69, 239, 434)-net over F₄, because

2 times m-reduction [i] would yield digital (69, 237, 434)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²³⁷, 434, F₄, 168) (dual of [434, 197, 169]-code), but
  - residual code [i] would yield OA(4⁶⁹, 265, S₄, 42), but
    - the linear programming bound shows that MÂ â‰¥ 52348 985175 743312 195164 699043 560255 242722 252419 549522 172173 323504 377479 561128 758476 800000 / 141682 475264 470819 104457 406563 443902 424070 593403 > 4⁶⁹ [i]

There is no (69, 239, 467)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 887857 393674 429091 833849 993016 010791 734744 281558 068070 767900 318879 334216 123255 210057 081180 361932 711682 762417 565644 364352 015990 169564 221017 990720 > 4²³⁹ [i]