Best Known (211, 211+43, s)-Nets in Base 2
(211, 211+43, 490)-Net over F2 — Constructive and digital
Digital (211, 254, 490)-net over F2, using
- 1 times m-reduction [i] based on digital (211, 255, 490)-net over F2, using
- trace code for nets [i] based on digital (7, 51, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 7 and N(F) ≥ 98, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- trace code for nets [i] based on digital (7, 51, 98)-net over F32, using
(211, 211+43, 1277)-Net over F2 — Digital
Digital (211, 254, 1277)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2254, 1277, F2, 3, 43) (dual of [(1277, 3), 3577, 44]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2254, 1369, F2, 3, 43) (dual of [(1369, 3), 3853, 44]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2254, 4107, F2, 43) (dual of [4107, 3853, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(2254, 4109, F2, 43) (dual of [4109, 3855, 44]-code), using
- construction X applied to Ce(42) ⊂ Ce(40) [i] based on
- linear OA(2253, 4096, F2, 43) (dual of [4096, 3843, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(2241, 4096, F2, 41) (dual of [4096, 3855, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(21, 13, F2, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(42) ⊂ Ce(40) [i] based on
- discarding factors / shortening the dual code based on linear OA(2254, 4109, F2, 43) (dual of [4109, 3855, 44]-code), using
- OOA 3-folding [i] based on linear OA(2254, 4107, F2, 43) (dual of [4107, 3853, 44]-code), using
- discarding factors / shortening the dual code based on linear OOA(2254, 1369, F2, 3, 43) (dual of [(1369, 3), 3853, 44]-NRT-code), using
(211, 211+43, 36713)-Net in Base 2 — Upper bound on s
There is no (211, 254, 36714)-net in base 2, because
- 1 times m-reduction [i] would yield (211, 253, 36714)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 14481 595582 233808 417646 335229 617480 822591 179101 371361 784768 249789 781392 403460 > 2253 [i]