Best Known (12, 12+79, s)-Nets in Base 25
(12, 12+79, 126)-Net over F25 — Constructive and digital
Digital (12, 91, 126)-net over F25, using
- t-expansion [i] based on digital (10, 91, 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+79, 1057)-Net over F25 — Upper bound on s (digital)
There is no digital (12, 91, 1058)-net over F25, because
- 1 times m-reduction [i] would yield digital (12, 90, 1058)-net over F25, but
- extracting embedded orthogonal array [i] would yield linear OA(2590, 1058, F25, 78) (dual of [1058, 968, 79]-code), but
- the Johnson bound shows that N ≤ 1570 250113 361094 143274 705222 136044 942385 000541 713135 864956 707159 896609 206550 529420 903037 073845 817203 700047 028410 162697 769624 370785 766070 610334 235785 074396 233524 729931 426667 994709 796398 865720 534030 293371 193874 615907 059516 123796 863069 317583 641573 034689 944981 953486 420562 058277 624199 113541 763030 914479 667955 119648 085767 380892 235557 935946 254855 572119 288531 011682 788019 284784 757934 874279 346047 549654 511465 974754 374353 815808 620850 339217 397251 364981 136292 145715 974618 900216 052725 930086 007902 443242 279938 464818 974558 086506 521481 934001 099784 780190 933481 113652 755516 286100 245184 179799 737890 055453 896945 003273 989214 400467 484215 248868 523761 764419 536721 981807 479463 948641 555824 074521 133364 635814 976437 002680 566697 932488 487352 646133 674105 333573 895703 647884 326760 447053 980687 982924 871437 209426 493718 312494 453446 221715 434504 009722 407510 886160 426516 202703 394076 694520 165815 935972 154682 914918 388452 595716 403596 186325 127560 751891 975367 409861 747558 645984 901020 304267 557908 414832 346357 128321 899841 943368 926962 870811 689189 641679 182823 729111 037871 741930 429192 854750 852892 797882 510880 689287 610364 192594 039563 122074 203165 736070 810107 830754 712543 006942 534420 100051 530372 591329 467288 373373 175833 895661 004609 402203 769524 908363 425839 986298 673777 721195 704026 063388 706819 958591 601489 064670 166879 744536 757416 740961 592277 476409 442788 976944 045168 027998 092833 999042 475501 020014 719084 559719 < 25968 [i]
- extracting embedded orthogonal array [i] would yield linear OA(2590, 1058, F25, 78) (dual of [1058, 968, 79]-code), but
(12, 12+79, 1058)-Net in Base 25 — Upper bound on s
There is no (12, 91, 1059)-net in base 25, because
- 1 times m-reduction [i] would yield (12, 90, 1059)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 653003 603352 508564 293415 991903 094814 008180 915002 474377 452172 570287 390422 436005 753256 564305 303225 560238 203040 220535 022616 847225 > 2590 [i]