Best Known (222−164, 222, s)-Nets in Base 4
(222−164, 222, 66)-Net over F4 — Constructive and digital
Digital (58, 222, 66)-net over F4, using
- t-expansion [i] based on digital (49, 222, 66)-net over F4, using
- net from sequence [i] based on digital (49, 65)-sequence over F4, using
- Niederreiter–Xing sequence construction 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]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 49 and N(F) ≥ 66, using
- net from sequence [i] based on digital (49, 65)-sequence over F4, using
(222−164, 222, 91)-Net over F4 — Digital
Digital (58, 222, 91)-net over F4, using
- t-expansion [i] based on digital (50, 222, 91)-net over F4, using
- net from sequence [i] based on digital (50, 90)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 50 and N(F) ≥ 91, using
- net from sequence [i] based on digital (50, 90)-sequence over F4, using
(222−164, 222, 274)-Net over F4 — Upper bound on s (digital)
There is no digital (58, 222, 275)-net over F4, because
- extracting embedded orthogonal array [i] would yield linear OA(4222, 275, F4, 164) (dual of [275, 53, 165]-code), but
- residual code [i] would yield OA(458, 110, S4, 41), but
- the linear programming bound shows that M ≥ 1232 155961 216230 168086 977678 056475 191089 409710 805770 829825 897828 417067 677196 009558 961106 027628 461497 995772 222254 044023 160230 949557 990006 504845 840607 997118 369373 847851 273455 403008 / 13446 810158 281327 760184 995983 919490 800441 646762 168115 151101 794360 608708 987818 154852 821096 884734 101312 394526 499211 581201 357552 019470 400051 102575 > 458 [i]
- residual code [i] would yield OA(458, 110, S4, 41), but
(222−164, 222, 381)-Net in Base 4 — Upper bound on s
There is no (58, 222, 382)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 50 954190 017816 636390 859898 435325 448928 515089 618972 717952 769801 909140 110956 240510 481231 026920 671395 412678 935238 162697 117629 972127 940330 > 4222 [i]