MinT - Best Known (115−76, 115, s)-Nets in Base 4

Best Known (115−76, 115, s)-Nets in Base 4

Digital (39, 115, 56)-net over F₄, using

t-expansion [i] based on digital (33, 115, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
    - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

Digital (39, 115, 66)-net over F₄, using

t-expansion [i] based on digital (37, 115, 66)-net over F₄, using
- net from sequence [i] based on digital (37, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 37 and N(F) ≥ 66, using
    - Drinfeld modules of rank 1 [i]

There is no (39, 115, 298)-net in base 4, because

1 times m-reduction [i] would yield (39, 114, 298)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹¹⁴, 298, S₄, 75), but
  - 2 times code embedding in larger space [i] would yield OA(4¹¹⁶, 300, S₄, 75), but
    - the linear programming bound shows that MÂ â‰¥ 6464 797858 730510 075007 646352 191601 758254 767824 883094 669028 865893 281872 177308 057446 516693 897945 662891 433844 560362 554677 851191 622996 365656 035845 716401 200310 839612 578911 449308 695601 901308 093236 233816 011409 843058 184451 226488 153734 498366 399468 218499 348610 322268 160000 000000 / 703971 314392 227904 434212 687438 485737 129519 187316 060725 351259 006871 197254 498335 242876 003553 867954 285607 077562 883304 888089 357569 168487 807481 727288 225812 136097 878623 386748 108103 111971 251088 305096 083141 > 4¹¹⁶ [i]