Best Known (135, 135+∞, s)-Nets in Base 4
(135, 135+∞, 130)-Net over F4 — Constructive and digital
Digital (135, m, 130)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (135, 129)-sequence over F4, using
- t-expansion [i] based on digital (105, 129)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 105 and N(F) ≥ 130, using
- T7 from the second tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 105 and N(F) ≥ 130, using
- t-expansion [i] based on digital (105, 129)-sequence over F4, using
(135, 135+∞, 176)-Net over F4 — Digital
Digital (135, m, 176)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (135, 175)-sequence over F4, using
- t-expansion [i] based on digital (125, 175)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 125 and N(F) ≥ 176, using
- t-expansion [i] based on digital (125, 175)-sequence over F4, using
(135, 135+∞, 421)-Net in Base 4 — Upper bound on s
There is no (135, m, 422)-net in base 4 for arbitrarily large m, because
- m-reduction [i] would yield (135, 2104, 422)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(42104, 422, S4, 5, 1969), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 5878 290385 126432 397762 964642 963925 833145 793328 325656 611238 520122 756921 582808 714624 027941 397960 664513 244210 798324 234285 210144 718160 079757 195493 400420 347249 801848 398025 401559 601041 693089 926536 772258 681131 211155 412421 128649 934864 521654 726723 338719 502672 017142 874523 329094 170816 835423 162734 031218 481830 987144 829254 045430 888828 812188 594252 120377 616659 536038 283971 486279 643064 773252 828672 777711 615579 753767 907498 745199 621298 899431 136380 865754 684386 897253 156036 125996 581254 043936 693307 073696 019395 769200 364699 846175 537374 452542 539579 555871 735673 269887 976611 658256 918278 127756 817183 648251 797351 882378 418879 133094 843924 490885 827329 943698 221803 808619 533103 906331 149829 086753 793086 886079 774505 347225 329611 882488 345053 104960 923490 278901 393281 777572 103758 748434 606629 485589 150588 060179 140975 592476 909024 013011 660488 336789 934845 401204 275960 221129 770660 600152 620848 351715 129718 478273 237581 904759 081813 360860 905059 539848 070745 628867 159559 909377 723451 800503 590085 801116 368519 171689 104237 148508 177647 190989 321869 392038 319598 483454 465924 585369 970013 770418 116784 394780 480690 169338 653497 677784 797003 768307 920411 355290 552870 752072 366318 735640 485036 065408 365851 551667 187929 676828 243840 384793 184790 910743 917732 738830 500702 057068 180811 658217 130156 085975 459377 204530 055247 621994 481284 922688 405504 / 985 > 42104 [i]
- extracting embedded OOA [i] would yield OOA(42104, 422, S4, 5, 1969), but