Best Known (157, 157+25, s)-Nets in Base 2
(157, 157+25, 2731)-Net over F2 — Constructive and digital
Digital (157, 182, 2731)-net over F2, using
- net defined by OOA [i] based on linear OOA(2182, 2731, F2, 25, 25) (dual of [(2731, 25), 68093, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2182, 32773, F2, 25) (dual of [32773, 32591, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2182, 32784, F2, 25) (dual of [32784, 32602, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(2181, 32768, F2, 25) (dual of [32768, 32587, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2166, 32768, F2, 23) (dual of [32768, 32602, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2182, 32784, F2, 25) (dual of [32784, 32602, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2182, 32773, F2, 25) (dual of [32773, 32591, 26]-code), using
(157, 157+25, 5464)-Net over F2 — Digital
Digital (157, 182, 5464)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2182, 5464, F2, 6, 25) (dual of [(5464, 6), 32602, 26]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2182, 32784, F2, 25) (dual of [32784, 32602, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(2181, 32768, F2, 25) (dual of [32768, 32587, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2166, 32768, F2, 23) (dual of [32768, 32602, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(24) ⊂ Ce(22) [i] based on
- OOA 6-folding [i] based on linear OA(2182, 32784, F2, 25) (dual of [32784, 32602, 26]-code), using
(157, 157+25, 183592)-Net in Base 2 — Upper bound on s
There is no (157, 182, 183593)-net in base 2, because
- 1 times m-reduction [i] would yield (157, 181, 183593)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 3 065019 018620 290204 964862 486332 598457 016622 367442 919512 > 2181 [i]