Best Known (191−27, 191, s)-Nets in Base 2
(191−27, 191, 1263)-Net over F2 — Constructive and digital
Digital (164, 191, 1263)-net over F2, using
- net defined by OOA [i] based on linear OOA(2191, 1263, F2, 27, 27) (dual of [(1263, 27), 33910, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2191, 16420, F2, 27) (dual of [16420, 16229, 28]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2190, 16419, F2, 27) (dual of [16419, 16229, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- linear OA(2183, 16384, F2, 27) (dual of [16384, 16201, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2155, 16384, F2, 23) (dual of [16384, 16229, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(27, 35, F2, 3) (dual of [35, 28, 4]-code or 35-cap in PG(6,2)), using
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 4 [i]
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2190, 16419, F2, 27) (dual of [16419, 16229, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2191, 16420, F2, 27) (dual of [16420, 16229, 28]-code), using
(191−27, 191, 3284)-Net over F2 — Digital
Digital (164, 191, 3284)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2191, 3284, F2, 5, 27) (dual of [(3284, 5), 16229, 28]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2191, 16420, F2, 27) (dual of [16420, 16229, 28]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2190, 16419, F2, 27) (dual of [16419, 16229, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- linear OA(2183, 16384, F2, 27) (dual of [16384, 16201, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2155, 16384, F2, 23) (dual of [16384, 16229, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(27, 35, F2, 3) (dual of [35, 28, 4]-code or 35-cap in PG(6,2)), using
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 4 [i]
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2190, 16419, F2, 27) (dual of [16419, 16229, 28]-code), using
- OOA 5-folding [i] based on linear OA(2191, 16420, F2, 27) (dual of [16420, 16229, 28]-code), using
(191−27, 191, 142238)-Net in Base 2 — Upper bound on s
There is no (164, 191, 142239)-net in base 2, because
- 1 times m-reduction [i] would yield (164, 190, 142239)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1569 338831 166271 548933 165115 477894 095918 269002 667794 182724 > 2190 [i]