MinT - Best Known (126−85, 126, s)-Nets in Base 4

Best Known (126−85, 126, s)-Nets in Base 4

Digital (41, 126, 56)-net over F₄, using

t-expansion [i] based on digital (33, 126, 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 (41, 126, 75)-net over F₄, using

t-expansion [i] based on digital (40, 126, 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 (41, 126, 297)-net in base 4, because

2 times m-reduction [i] would yield (41, 124, 297)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹²⁴, 297, S₄, 83), but
  - 3 times code embedding in larger space [i] would yield OA(4¹²⁷, 300, S₄, 83), but
    - the linear programming bound shows that MÂ â‰¥ 23 469642 541307 124705 404222 022984 144794 721391 122290 751092 642846 233129 801567 843921 331513 551043 264355 934481 703128 191262 686702 008695 646908 644783 998160 262004 107351 089756 851243 202033 203868 207899 878664 300563 379107 391462 560767 818388 006329 884983 992307 349876 630804 307146 722579 073895 153807 583488 245753 130826 740574 907907 334153 286778 265284 969619 618654 253378 727534 466879 681442 400358 594005 177975 182454 366894 886858 805609 336320 757192 231076 495360 / 626 892930 408837 560750 056226 482987 077460 731023 530653 628449 955197 841352 112635 445051 238446 358741 996695 770467 206838 063116 195904 382671 599291 538587 216473 435432 498925 325120 828016 957414 868969 359071 732814 334180 291076 512917 341578 864782 770104 902159 061540 953718 928993 902451 599106 470097 391087 715946 096566 378935 380788 001637 608652 595498 637124 865798 715685 636579 197801 > 4¹²⁷ [i]