Best Known (228, 228+31, s)-Nets in Base 2
(228, 228+31, 8739)-Net over F2 — Constructive and digital
Digital (228, 259, 8739)-net over F2, using
- 22 times duplication [i] based on digital (226, 257, 8739)-net over F2, using
- net defined by OOA [i] based on linear OOA(2257, 8739, F2, 31, 31) (dual of [(8739, 31), 270652, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2257, 131086, F2, 31) (dual of [131086, 130829, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2257, 131090, F2, 31) (dual of [131090, 130833, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(2256, 131072, F2, 31) (dual of [131072, 130816, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- 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(21, 18, F2, 1) (dual of [18, 17, 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(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(2257, 131090, F2, 31) (dual of [131090, 130833, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2257, 131086, F2, 31) (dual of [131086, 130829, 32]-code), using
- net defined by OOA [i] based on linear OOA(2257, 8739, F2, 31, 31) (dual of [(8739, 31), 270652, 32]-NRT-code), using
(228, 228+31, 18703)-Net over F2 — Digital
Digital (228, 259, 18703)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2259, 18703, F2, 7, 31) (dual of [(18703, 7), 130662, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2259, 18727, F2, 7, 31) (dual of [(18727, 7), 130830, 32]-NRT-code), using
- 22 times duplication [i] based on linear OOA(2257, 18727, F2, 7, 31) (dual of [(18727, 7), 130832, 32]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2257, 131089, F2, 31) (dual of [131089, 130832, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2257, 131090, F2, 31) (dual of [131090, 130833, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(2256, 131072, F2, 31) (dual of [131072, 130816, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- 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(21, 18, F2, 1) (dual of [18, 17, 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(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(2257, 131090, F2, 31) (dual of [131090, 130833, 32]-code), using
- OOA 7-folding [i] based on linear OA(2257, 131089, F2, 31) (dual of [131089, 130832, 32]-code), using
- 22 times duplication [i] based on linear OOA(2257, 18727, F2, 7, 31) (dual of [(18727, 7), 130832, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2259, 18727, F2, 7, 31) (dual of [(18727, 7), 130830, 32]-NRT-code), using
(228, 228+31, 967102)-Net in Base 2 — Upper bound on s
There is no (228, 259, 967103)-net in base 2, because
- 1 times m-reduction [i] would yield (228, 258, 967103)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 463169 054459 161615 080989 674454 903085 318628 343742 716421 672671 710152 469921 575618 > 2258 [i]