Best Known (136, 159, s)-Nets in Base 2
(136, 159, 1491)-Net over F2 — Constructive and digital
Digital (136, 159, 1491)-net over F2, using
- net defined by OOA [i] based on linear OOA(2159, 1491, F2, 23, 23) (dual of [(1491, 23), 34134, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2159, 16402, F2, 23) (dual of [16402, 16243, 24]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2156, 16399, F2, 23) (dual of [16399, 16243, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- 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(2141, 16384, F2, 21) (dual of [16384, 16243, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(21, 15, F2, 1) (dual of [15, 14, 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(22) ⊂ Ce(20) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(2156, 16399, F2, 23) (dual of [16399, 16243, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2159, 16402, F2, 23) (dual of [16402, 16243, 24]-code), using
(136, 159, 3280)-Net over F2 — Digital
Digital (136, 159, 3280)-net over F2, using
- 22 times duplication [i] based on digital (134, 157, 3280)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2157, 3280, F2, 5, 23) (dual of [(3280, 5), 16243, 24]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2157, 16400, F2, 23) (dual of [16400, 16243, 24]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2156, 16399, F2, 23) (dual of [16399, 16243, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- 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(2141, 16384, F2, 21) (dual of [16384, 16243, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(21, 15, F2, 1) (dual of [15, 14, 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(22) ⊂ Ce(20) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2156, 16399, F2, 23) (dual of [16399, 16243, 24]-code), using
- OOA 5-folding [i] based on linear OA(2157, 16400, F2, 23) (dual of [16400, 16243, 24]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2157, 3280, F2, 5, 23) (dual of [(3280, 5), 16243, 24]-NRT-code), using
(136, 159, 103474)-Net in Base 2 — Upper bound on s
There is no (136, 159, 103475)-net in base 2, because
- 1 times m-reduction [i] would yield (136, 158, 103475)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 365403 453990 538534 622430 163042 562096 771414 036076 > 2158 [i]