Best Known (65, 65+121, s)-Nets in Base 3
(65, 65+121, 48)-Net over F3 — Constructive and digital
Digital (65, 186, 48)-net over F3, using
- t-expansion [i] based on digital (45, 186, 48)-net over F3, using
- net from sequence [i] based on digital (45, 47)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 45 and N(F) ≥ 48, using
- net from sequence [i] based on digital (45, 47)-sequence over F3, using
(65, 65+121, 64)-Net over F3 — Digital
Digital (65, 186, 64)-net over F3, using
- t-expansion [i] based on digital (49, 186, 64)-net over F3, using
- net from sequence [i] based on digital (49, 63)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 49 and N(F) ≥ 64, using
- net from sequence [i] based on digital (49, 63)-sequence over F3, using
(65, 65+121, 236)-Net over F3 — Upper bound on s (digital)
There is no digital (65, 186, 237)-net over F3, because
- 1 times m-reduction [i] would yield digital (65, 185, 237)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3185, 237, F3, 120) (dual of [237, 52, 121]-code), but
- residual code [i] would yield OA(365, 116, S3, 40), but
- the linear programming bound shows that M ≥ 1010 233715 716651 107612 542215 638091 218609 672761 165524 903593 924473 005111 306769 562821 186134 762126 385755 152296 422613 586112 342241 377727 269019 414533 549609 685737 803087 / 94 423144 003278 216048 118473 500831 454684 060079 864523 188758 682434 278034 186489 210172 080884 936271 279930 533580 650501 446289 752944 864745 > 365 [i]
- residual code [i] would yield OA(365, 116, S3, 40), but
- extracting embedded orthogonal array [i] would yield linear OA(3185, 237, F3, 120) (dual of [237, 52, 121]-code), but
(65, 65+121, 287)-Net in Base 3 — Upper bound on s
There is no (65, 186, 288)-net in base 3, because
- 1 times m-reduction [i] would yield (65, 185, 288)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 20077 063391 851850 709262 084054 297405 381375 874272 029976 457593 895135 148262 723909 524099 382017 > 3185 [i]