Best Known (78−68, 78, s)-Nets in Base 25
(78−68, 78, 126)-Net over F25 — Constructive and digital
Digital (10, 78, 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
(78−68, 78, 889)-Net over F25 — Upper bound on s (digital)
There is no digital (10, 78, 890)-net over F25, because
- extracting embedded orthogonal array [i] would yield linear OA(2578, 890, F25, 68) (dual of [890, 812, 69]-code), but
- the Johnson bound shows that N ≤ 13 376365 847166 970327 653036 804114 383206 117163 077246 137696 034975 858399 417557 210016 368309 414633 229640 408057 577353 601076 605999 187805 049791 007743 013590 196549 057998 844522 147162 525097 792093 656731 358373 741677 941850 639132 800991 586716 487656 938531 557923 447134 237642 875133 957597 860532 951949 535124 460320 022540 014264 558253 157484 531641 311401 746080 793827 400943 638166 684427 263148 338662 542989 017196 584608 244516 143511 557402 119416 786635 749894 317068 605069 462988 698676 112575 849068 337067 155639 342437 305787 403949 334182 153926 370742 468864 021880 991051 848865 077837 889213 595397 414822 711306 628498 954456 691500 274694 409311 875420 378353 763311 390088 470976 825685 204822 820129 793454 635847 971815 303404 833574 005688 517283 236408 594267 449446 342076 066970 227812 152801 042712 078853 782130 596707 599690 682477 892388 533118 098827 078599 058194 417359 059200 383226 937868 793738 856027 356635 170784 599035 627261 299480 707631 248884 993675 496750 077222 732304 839868 081237 271101 891491 380477 611561 315075 948640 551231 061305 815401 180230 131443 385719 763583 162578 839583 260597 933808 565076 918692 566014 545610 668465 235511 384775 017365 782652 666532 989256 922106 756967 741473 277250 048683 349605 041474 814654 035567 607910 257136 < 25812 [i]
(78−68, 78, 890)-Net in Base 25 — Upper bound on s
There is no (10, 78, 891)-net in base 25, because
- the generalized Rao bound for nets shows that 25m ≥ 11 105696 010250 995912 304422 526109 279584 281652 848866 326575 437724 512275 998181 621514 432968 576325 056579 095736 481425 > 2578 [i]