Best Known (12, 12+77, s)-Nets in Base 25
(12, 12+77, 126)-Net over F25 — Constructive and digital
Digital (12, 89, 126)-net over F25, using
- t-expansion [i] based on digital (10, 89, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
(12, 12+77, 1059)-Net over F25 — Upper bound on s (digital)
There is no digital (12, 89, 1060)-net over F25, because
- 1 times m-reduction [i] would yield digital (12, 88, 1060)-net over F25, but
- extracting embedded orthogonal array [i] would yield linear OA(2588, 1060, F25, 76) (dual of [1060, 972, 77]-code), but
- the Johnson bound shows that N ≤ 627 229819 591289 268340 451269 692196 152334 544567 870239 791793 035872 259917 480168 883251 685660 979055 448331 863484 400953 336672 243287 866087 182161 433622 538673 681862 300004 449937 535043 018079 688526 350967 509462 471232 005794 251766 192827 017449 058984 353628 879186 596478 665162 080003 125589 178056 283499 092908 319178 439707 885700 071359 198840 437059 169623 749688 740344 859936 514906 821134 151032 242483 686433 191459 428112 422822 815890 040294 854350 456028 370916 100994 307017 134482 720153 019722 585255 934468 285584 898323 971944 495324 548898 722326 800125 286795 591563 015720 866875 431333 066523 446495 330150 062815 759589 419132 128547 842088 403623 551340 909611 453397 590945 620926 673364 308788 515616 160929 070816 771755 133833 713344 948731 834708 012224 893714 453107 848727 732069 396406 029568 306618 983412 126894 332836 948035 508026 036714 093491 747086 650608 427257 413153 402778 241535 153201 867320 719716 737253 579730 604928 173564 668099 350217 367067 728029 456192 572815 278219 343175 974485 208302 491108 918925 115210 831314 599470 843339 363717 729012 854863 385691 393758 912569 679930 868813 972143 746868 049133 097752 319826 610931 485719 124782 140291 867732 591564 385135 046905 200004 958570 974469 224451 820177 542547 559967 040816 514535 044240 199351 313139 394686 806892 652456 385915 367699 943849 219883 761372 853119 227864 283405 477994 991639 721831 733458 257048 563900 855020 911528 056454 316964 918564 098973 700635 732300 777827 590063 810229 788823 190311 244946 486198 484767 454436 592102 133166 < 25972 [i]
- extracting embedded orthogonal array [i] would yield linear OA(2588, 1060, F25, 76) (dual of [1060, 972, 77]-code), but
(12, 12+77, 1061)-Net in Base 25 — Upper bound on s
There is no (12, 89, 1062)-net in base 25, because
- 1 times m-reduction [i] would yield (12, 88, 1062)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 1071 867426 732964 310015 732890 561769 581388 715409 827859 037330 739540 206930 858009 599196 771931 035695 063001 738379 603891 884558 881825 > 2588 [i]