Best Known (251−28, 251, s)-Nets in Base 2
(251−28, 251, 9366)-Net over F2 — Constructive and digital
Digital (223, 251, 9366)-net over F2, using
- net defined by OOA [i] based on linear OOA(2251, 9366, F2, 28, 28) (dual of [(9366, 28), 261997, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(2251, 131124, F2, 28) (dual of [131124, 130873, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(2251, 131135, F2, 28) (dual of [131135, 130884, 29]-code), using
- 1 times truncation [i] based on linear OA(2252, 131136, F2, 29) (dual of [131136, 130884, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- linear OA(2239, 131072, F2, 29) (dual of [131072, 130833, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(2188, 131072, F2, 23) (dual of [131072, 130884, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(213, 64, F2, 5) (dual of [64, 51, 6]-code), using
- an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- 1 times truncation [i] based on linear OA(2252, 131136, F2, 29) (dual of [131136, 130884, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2251, 131135, F2, 28) (dual of [131135, 130884, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(2251, 131124, F2, 28) (dual of [131124, 130873, 29]-code), using
(251−28, 251, 21855)-Net over F2 — Digital
Digital (223, 251, 21855)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2251, 21855, F2, 6, 28) (dual of [(21855, 6), 130879, 29]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2251, 131130, F2, 28) (dual of [131130, 130879, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(2251, 131135, F2, 28) (dual of [131135, 130884, 29]-code), using
- 1 times truncation [i] based on linear OA(2252, 131136, F2, 29) (dual of [131136, 130884, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- linear OA(2239, 131072, F2, 29) (dual of [131072, 130833, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(2188, 131072, F2, 23) (dual of [131072, 130884, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(213, 64, F2, 5) (dual of [64, 51, 6]-code), using
- an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- 1 times truncation [i] based on linear OA(2252, 131136, F2, 29) (dual of [131136, 130884, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2251, 131135, F2, 28) (dual of [131135, 130884, 29]-code), using
- OOA 6-folding [i] based on linear OA(2251, 131130, F2, 28) (dual of [131130, 130879, 29]-code), using
(251−28, 251, 1508306)-Net in Base 2 — Upper bound on s
There is no (223, 251, 1508307)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 3618 517993 888253 491305 760095 102976 812268 141639 766677 784959 730535 205329 923124 > 2251 [i]