Best Known (117, 206, s)-Nets in Base 2
(117, 206, 62)-Net over F2 — Constructive and digital
Digital (117, 206, 62)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (19, 63, 20)-net over F2, using
- net from sequence [i] based on digital (19, 19)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 19 and N(F) ≥ 20, using
- net from sequence [i] based on digital (19, 19)-sequence over F2, using
- digital (54, 143, 42)-net over F2, using
- net from sequence [i] based on digital (54, 41)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 54 and N(F) ≥ 42, using
- net from sequence [i] based on digital (54, 41)-sequence over F2, using
- digital (19, 63, 20)-net over F2, using
(117, 206, 79)-Net over F2 — Digital
Digital (117, 206, 79)-net over F2, using
(117, 206, 362)-Net over F2 — Upper bound on s (digital)
There is no digital (117, 206, 363)-net over F2, because
- 1 times m-reduction [i] would yield digital (117, 205, 363)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2205, 363, F2, 88) (dual of [363, 158, 89]-code), but
- construction Y1 [i] would yield
- OA(2204, 299, S2, 88), but
- adding a parity check bit [i] would yield OA(2205, 300, S2, 89), but
- the linear programming bound shows that M ≥ 416927 463188 475542 795463 940731 898983 181308 104641 213119 598295 609827 594303 154474 074874 094714 331435 368448 / 5584 401256 756961 297780 513782 298869 514125 > 2205 [i]
- adding a parity check bit [i] would yield OA(2205, 300, S2, 89), but
- linear OA(2158, 363, F2, 64) (dual of [363, 205, 65]-code), but
- the improved Johnson bound shows that N ≤ 1214 271568 452769 866939 025066 626311 211727 927188 642255 823141 634304 < 2205 [i]
- OA(2204, 299, S2, 88), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(2205, 363, F2, 88) (dual of [363, 158, 89]-code), but
(117, 206, 374)-Net in Base 2 — Upper bound on s
There is no (117, 206, 375)-net in base 2, because
- 1 times m-reduction [i] would yield (117, 205, 375)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 53 430801 671686 355025 890169 121466 558963 919592 764411 474887 620566 > 2205 [i]