Best Known (119, 119+91, s)-Nets in Base 2
(119, 119+91, 62)-Net over F2 — Constructive and digital
Digital (119, 210, 62)-net over F2, using
- 1 times m-reduction [i] based on digital (119, 211, 62)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (19, 65, 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, 146, 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, 65, 20)-net over F2, using
- (u, u+v)-construction [i] based on
(119, 119+91, 80)-Net over F2 — Digital
Digital (119, 210, 80)-net over F2, using
(119, 119+91, 358)-Net over F2 — Upper bound on s (digital)
There is no digital (119, 210, 359)-net over F2, because
- 1 times m-reduction [i] would yield digital (119, 209, 359)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2209, 359, F2, 90) (dual of [359, 150, 91]-code), but
- construction Y1 [i] would yield
- OA(2208, 299, S2, 90), but
- adding a parity check bit [i] would yield OA(2209, 300, S2, 91), but
- the linear programming bound shows that M ≥ 437987 836379 661851 428112 482969 460104 960459 291237 979833 080590 515802 667419 590325 215760 903409 709975 863296 / 433 753365 263741 276278 643266 060465 678125 > 2209 [i]
- adding a parity check bit [i] would yield OA(2209, 300, S2, 91), but
- linear OA(2150, 359, F2, 60) (dual of [359, 209, 61]-code), but
- discarding factors / shortening the dual code would yield linear OA(2150, 354, F2, 60) (dual of [354, 204, 61]-code), but
- the improved Johnson bound shows that N ≤ 570 693273 077464 953259 984159 736256 114541 311702 095039 925212 853150 < 2204 [i]
- discarding factors / shortening the dual code would yield linear OA(2150, 354, F2, 60) (dual of [354, 204, 61]-code), but
- OA(2208, 299, S2, 90), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(2209, 359, F2, 90) (dual of [359, 150, 91]-code), but
(119, 119+91, 378)-Net in Base 2 — Upper bound on s
There is no (119, 210, 379)-net in base 2, because
- 1 times m-reduction [i] would yield (119, 209, 379)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 897 750651 213333 649356 162003 931970 929177 531355 213656 732703 603152 > 2209 [i]