MinT - Best Known (51, 51+159, s)-Nets in Base 4

Best Known (51, 51+159, s)-Nets in Base 4

(51, 51+159, 66)-Net over F₄ — Constructive and digital

Digital (51, 210, 66)-net over F₄, using

t-expansion [i] based on digital (49, 210, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(51, 51+159, 91)-Net over F₄ — Digital

Digital (51, 210, 91)-net over F₄, using

t-expansion [i] based on digital (50, 210, 91)-net over F₄, using
- net from sequence [i] based on digital (50, 90)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 50 and N(F) ≥ 91, using
    - a curve from the manYPoints database [i]

(51, 51+159, 214)-Net over F₄ — Upper bound on s (digital)

There is no digital (51, 210, 215)-net over F₄, because

3 times m-reduction [i] would yield digital (51, 207, 215)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁰⁷, 215, F₄, 156) (dual of [215, 8, 157]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(4²⁰⁶, 211, F₄, 156) (dual of [211, 5, 157]-code), but
      - residual code [i] would yield linear OA(4⁵⁰, 54, F₄, 39) (dual of [54, 4, 40]-code), but
        3 times truncation [i] would yield linear OA(4⁴⁷, 51, F₄, 36) (dual of [51, 4, 37]-code), but
        a result by Landjev, Maruta, and Hill [i]
    2. OA(4⁸, 215, S₄, 4), but
      - discarding factors would yield OA(4⁸, 121, S₄, 4), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 65704 > 4⁸ [i]

(51, 51+159, 296)-Net in Base 4 — Upper bound on s

There is no (51, 210, 297)-net in base 4, because

50 times m-reduction [i] would yield (51, 160, 297)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4¹⁶⁰, 297, S₄, 109), but
  - 3 times code embedding in larger space [i] would yield OA(4¹⁶³, 300, S₄, 109), but
    - the linear programming bound shows that MÂ â‰¥ 8 464628 620685 303799 641532 702091 648229 306306 018905 789529 524291 355322 907050 775490 485015 472797 926481 512459 111645 061846 121711 271262 485718 789063 277920 642870 575525 965715 473112 303555 905333 205605 386730 426241 764666 473440 037313 207471 384293 998241 638361 590587 784985 333805 211326 107965 662310 596475 534832 743164 666681 710435 385827 242945 735282 806533 359251 451373 682496 947126 249928 010306 759678 464627 244345 575141 502578 293166 452201 656662 745544 187910 102207 124893 914791 852222 399372 582590 940059 694271 963942 843967 307519 271729 211883 132520 499405 503460 690663 546106 254632 847756 563871 678160 926784 538665 864722 013378 367220 541645 343747 080032 801040 473128 984728 287622 005132 676143 769268 182113 404481 574415 357033 841665 346615 737557 681028 068460 307746 390016 / 58507 773896 969173 286262 004882 425009 217647 370320 705564 699864 967449 325045 255104 627416 234082 746442 427784 881046 418017 053542 895824 632469 013989 930358 814771 742000 630486 409427 709699 986375 771286 038746 569037 585403 288576 489419 064021 553229 427653 059477 355994 492045 140445 835038 050400 407744 351141 166731 645775 405880 298749 106576 452336 500949 515816 312820 547873 019373 515697 296633 040866 524679 340051 766785 242210 065144 551669 611353 000016 122330 320952 751635 405353 975165 600537 815777 260236 441170 624710 555045 295395 973417 233997 679742 229407 872510 882226 515661 153603 723506 086762 064908 948089 824582 619268 759820 263783 024711 275252 152307 556197 369539 309969 > 4¹⁶³ [i]

Best Known (51, 51+159, s)-Nets in Base 4

(51, 51+159, 66)-Net over F4 — Constructive and digital

(51, 51+159, 91)-Net over F4 — Digital

(51, 51+159, 214)-Net over F4 — Upper bound on s (digital)

(51, 51+159, 296)-Net in Base 4 — Upper bound on s

(51, 51+159, 66)-Net over F₄ — Constructive and digital

(51, 51+159, 91)-Net over F₄ — Digital

(51, 51+159, 214)-Net over F₄ — Upper bound on s (digital)