MinT - Best Known (133−89, 133, s)-Nets in Base 4

Best Known (133−89, 133, s)-Nets in Base 4

(133−89, 133, 56)-Net over F₄ — Constructive and digital

Digital (44, 133, 56)-net over F₄, using

t-expansion [i] based on digital (33, 133, 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]

(133−89, 133, 75)-Net over F₄ — Digital

Digital (44, 133, 75)-net over F₄, using

t-expansion [i] based on digital (40, 133, 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]

(133−89, 133, 297)-Net in Base 4 — Upper bound on s

There is no (44, 133, 298)-net in base 4, because

1 times m-reduction [i] would yield (44, 132, 298)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹³², 298, S₄, 88), but
  - 2 times code embedding in larger space [i] would yield OA(4¹³⁴, 300, S₄, 88), but
    - the linear programming bound shows that MÂ â‰¥ 5064 666519 196273 720359 957160 453531 439292 263994 048519 181650 217094 053152 090401 577550 580614 771321 755805 921895 812862 361787 400896 027536 234706 584830 243715 121125 378166 129527 230569 230623 066946 357421 032821 558304 963930 973841 794224 966336 207917 208968 708013 869798 547880 195312 957940 530209 124856 506843 528583 849889 153654 612021 526496 806290 579288 964514 854847 589042 988738 974236 189862 325201 017048 247926 072986 379740 528091 671902 097410 687862 943138 661367 388402 043801 402980 042847 808544 422009 093495 088415 147507 514552 520966 726022 652111 714160 137509 760789 435009 222813 746922 550436 792337 694720 000000 / 10 319227 060254 222010 640912 950222 869938 668873 053245 195420 643581 448207 956929 016749 592690 007745 395132 545138 525468 361150 566962 170541 590239 667564 802060 210579 590568 813476 314579 524728 420027 120221 193332 015849 649418 580053 918218 680562 260695 030701 553173 940153 895392 233393 110834 818623 398878 373923 781987 444136 842707 148741 774765 920050 535321 129918 813372 172481 253534 670503 246711 566591 628458 776629 882041 191569 173261 753083 461670 327890 946532 135250 749688 245715 217284 467235 619374 068972 642949 544751 083531 432777 > 4¹³⁴ [i]

Best Known (133−89, 133, s)-Nets in Base 4

(133−89, 133, 56)-Net over F4 — Constructive and digital

(133−89, 133, 75)-Net over F4 — Digital

(133−89, 133, 297)-Net in Base 4 — Upper bound on s

(133−89, 133, 56)-Net over F₄ — Constructive and digital

(133−89, 133, 75)-Net over F₄ — Digital