Best Known (68, 68+188, s)-Nets in Base 4
(68, 68+188, 66)-Net over F4 — Constructive and digital
Digital (68, 256, 66)-net over F4, using
- t-expansion [i] based on digital (49, 256, 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
(68, 68+188, 99)-Net over F4 — Digital
Digital (68, 256, 99)-net over F4, using
- t-expansion [i] based on digital (61, 256, 99)-net over F4, using
- net from sequence [i] based on digital (61, 98)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 61 and N(F) ≥ 99, using
- net from sequence [i] based on digital (61, 98)-sequence over F4, using
(68, 68+188, 328)-Net over F4 — Upper bound on s (digital)
There is no digital (68, 256, 329)-net over F4, because
- extracting embedded orthogonal array [i] would yield linear OA(4256, 329, F4, 188) (dual of [329, 73, 189]-code), but
- residual code [i] would yield OA(468, 140, S4, 47), but
- the linear programming bound shows that M ≥ 4 654819 034028 581238 257287 026361 830986 068286 283057 495782 040750 182767 226084 887911 211399 130524 177049 365476 187506 819783 810261 868064 094792 489370 047724 354082 523054 068745 865536 810824 821096 908875 921495 293874 417204 028277 391129 766840 473604 360300 402346 099890 643466 230695 991515 072657 327908 774115 015089 759021 885447 095951 554698 978304 983040 / 52 356806 020996 136503 048528 999171 798309 546097 784055 794841 348033 950332 913120 624137 419013 472408 780999 008490 616180 475616 157784 140759 720487 396688 307273 081754 874604 372029 182708 524635 237827 260260 184676 647940 453297 930054 409979 764567 296134 969415 875473 293271 581970 074707 039796 728818 821048 930327 > 468 [i]
- residual code [i] would yield OA(468, 140, S4, 47), but
(68, 68+188, 446)-Net in Base 4 — Upper bound on s
There is no (68, 256, 447)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 14911 682160 902094 883968 841416 340536 212713 962103 734722 371045 785413 676873 226195 723709 550897 764125 331018 283848 232523 354164 490817 570884 985173 436922 391074 672010 > 4256 [i]