MinT - Best Known (129−87, 129, s)-Nets in Base 4

Best Known (129−87, 129, s)-Nets in Base 4

Digital (42, 129, 56)-net over F₄, using

t-expansion [i] based on digital (33, 129, 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 (42, 129, 75)-net over F₄, using

t-expansion [i] based on digital (40, 129, 75)-net over F₄, using
- net from sequence [i] based on digital (40, 74)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 40 and N(F) ≥ 75, using
    - Drinfeld modules of rank 1 [i]

There is no (42, 129, 297)-net in base 4, because

1 times m-reduction [i] would yield (42, 128, 297)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹²⁸, 297, S₄, 86), but
  - 3 times code embedding in larger space [i] would yield OA(4¹³¹, 300, S₄, 86), but
    - the linear programming bound shows that MÂ â‰¥ 138 416617 853052 152347 536798 798654 835087 020376 222814 713214 141827 603142 133611 729006 059680 594468 051739 330837 251844 737246 324435 770942 954643 084128 481319 607223 742827 155949 327327 309844 342768 845789 945402 484876 490851 097309 421106 691670 383746 811200 264843 343791 326307 599073 770776 041729 944010 948259 605594 133654 043319 021066 631403 029985 895720 449461 316044 853428 913689 893237 241319 241644 134824 192510 074759 334206 234926 170022 820549 570998 450788 281685 186306 508936 955123 968929 366119 807536 891873 846062 089718 890055 625594 634240 / 12 827996 571606 367196 603122 092368 604438 519009 544353 512189 994213 726836 092060 891296 935301 166803 945681 970990 301549 273696 560280 498093 157138 734296 028074 438602 433000 808251 262825 045358 718240 895151 171144 795357 947930 640334 734373 765805 529958 790312 420427 325490 736197 048748 105679 560680 716273 124551 086061 156158 136900 549163 796460 475427 451646 653959 137287 976801 572547 437148 165790 265394 503962 295957 942800 530036 287951 901079 825595 943331 476449 > 4¹³¹ [i]