Best Known (248−43, 248, s)-Nets in Base 2
(248−43, 248, 380)-Net over F2 — Constructive and digital
Digital (205, 248, 380)-net over F2, using
- 2 times m-reduction [i] based on digital (205, 250, 380)-net over F2, using
- trace code for nets [i] based on digital (5, 50, 76)-net over F32, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 5 and N(F) ≥ 76, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- trace code for nets [i] based on digital (5, 50, 76)-net over F32, using
(248−43, 248, 1048)-Net over F2 — Digital
Digital (205, 248, 1048)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2248, 1048, F2, 2, 43) (dual of [(1048, 2), 1848, 44]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2247, 1048, F2, 2, 43) (dual of [(1048, 2), 1849, 44]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2245, 1047, F2, 2, 43) (dual of [(1047, 2), 1849, 44]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2245, 2094, F2, 43) (dual of [2094, 1849, 44]-code), using
- construction X applied to Ce(42) ⊂ Ce(36) [i] based on
- linear OA(2232, 2048, F2, 43) (dual of [2048, 1816, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(2199, 2048, F2, 37) (dual of [2048, 1849, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(213, 46, F2, 5) (dual of [46, 33, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(213, 63, F2, 5) (dual of [63, 50, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(213, 63, F2, 5) (dual of [63, 50, 6]-code), using
- construction X applied to Ce(42) ⊂ Ce(36) [i] based on
- OOA 2-folding [i] based on linear OA(2245, 2094, F2, 43) (dual of [2094, 1849, 44]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2245, 1047, F2, 2, 43) (dual of [(1047, 2), 1849, 44]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2247, 1048, F2, 2, 43) (dual of [(1048, 2), 1849, 44]-NRT-code), using
(248−43, 248, 30111)-Net in Base 2 — Upper bound on s
There is no (205, 248, 30112)-net in base 2, because
- 1 times m-reduction [i] would yield (205, 247, 30112)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 226 239085 013618 005768 059514 545006 217412 232269 700752 808499 561675 381179 447667 > 2247 [i]