Best Known (135, 135+21, s)-Nets in Base 2
(135, 135+21, 3278)-Net over F2 — Constructive and digital
Digital (135, 156, 3278)-net over F2, using
- 24 times duplication [i] based on digital (131, 152, 3278)-net over F2, using
- net defined by OOA [i] based on linear OOA(2152, 3278, F2, 21, 21) (dual of [(3278, 21), 68686, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2152, 32781, F2, 21) (dual of [32781, 32629, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2152, 32784, F2, 21) (dual of [32784, 32632, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(2151, 32768, F2, 21) (dual of [32768, 32617, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2136, 32768, F2, 19) (dual of [32768, 32632, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(21, 16, F2, 1) (dual of [16, 15, 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(20) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(2152, 32784, F2, 21) (dual of [32784, 32632, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2152, 32781, F2, 21) (dual of [32781, 32629, 22]-code), using
- net defined by OOA [i] based on linear OOA(2152, 3278, F2, 21, 21) (dual of [(3278, 21), 68686, 22]-NRT-code), using
(135, 135+21, 6557)-Net over F2 — Digital
Digital (135, 156, 6557)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2156, 6557, F2, 5, 21) (dual of [(6557, 5), 32629, 22]-NRT-code), using
- 23 times duplication [i] based on linear OOA(2153, 6557, F2, 5, 21) (dual of [(6557, 5), 32632, 22]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2153, 32785, F2, 21) (dual of [32785, 32632, 22]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2152, 32784, F2, 21) (dual of [32784, 32632, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(2151, 32768, F2, 21) (dual of [32768, 32617, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2136, 32768, F2, 19) (dual of [32768, 32632, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(21, 16, F2, 1) (dual of [16, 15, 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(20) ⊂ Ce(18) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2152, 32784, F2, 21) (dual of [32784, 32632, 22]-code), using
- OOA 5-folding [i] based on linear OA(2153, 32785, F2, 21) (dual of [32785, 32632, 22]-code), using
- 23 times duplication [i] based on linear OOA(2153, 6557, F2, 5, 21) (dual of [(6557, 5), 32632, 22]-NRT-code), using
(135, 135+21, 209851)-Net in Base 2 — Upper bound on s
There is no (135, 156, 209852)-net in base 2, because
- 1 times m-reduction [i] would yield (135, 155, 209852)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 45673 906150 621502 951530 650115 798641 217239 112689 > 2155 [i]