MinT - Best Known (117−78, 117, s)-Nets in Base 4

Best Known (117−78, 117, s)-Nets in Base 4

Digital (39, 117, 56)-net over F₄, using

t-expansion [i] based on digital (33, 117, 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, 117, 66)-net over F₄, using

t-expansion [i] based on digital (37, 117, 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, 117, 297)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4¹¹⁷, 297, S₄, 78), but
- 3 times code embedding in larger space [i] would yield OA(4¹²⁰, 300, S₄, 78), but
  - the linear programming bound shows that MÂ â‰¥ 27785 634061 051015 445205 539705 128243 831042 551761 765354 054520 845232 732470 134314 199734 282257 427758 804287 043032 205850 749893 192165 896354 157338 298560 633749 205335 186265 302016 421228 031242 873449 666046 061013 333606 281390 970484 324358 168639 942899 000832 501013 168206 554396 066848 135260 716578 430241 923947 265270 227017 587825 523431 792579 706880 / 10286 735798 548365 617851 451547 986060 045782 385628 992778 983314 728378 150249 898739 975669 890309 884984 154267 733627 699774 486308 909201 945709 402212 881835 886252 018050 944325 637391 617180 508067 045795 228375 309453 655675 924503 954280 909191 453936 867965 130048 192574 132353 > 4¹²⁰ [i]