Best Known (256−34, 256, s)-Nets in Base 2
(256−34, 256, 1928)-Net over F2 — Constructive and digital
Digital (222, 256, 1928)-net over F2, using
- net defined by OOA [i] based on linear OOA(2256, 1928, F2, 34, 34) (dual of [(1928, 34), 65296, 35]-NRT-code), using
- OA 17-folding and stacking [i] based on linear OA(2256, 32776, F2, 34) (dual of [32776, 32520, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(2256, 32783, F2, 34) (dual of [32783, 32527, 35]-code), using
- 1 times truncation [i] based on linear OA(2257, 32784, F2, 35) (dual of [32784, 32527, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(32) [i] based on
- linear OA(2256, 32768, F2, 35) (dual of [32768, 32512, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2241, 32768, F2, 33) (dual of [32768, 32527, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [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(34) ⊂ Ce(32) [i] based on
- 1 times truncation [i] based on linear OA(2257, 32784, F2, 35) (dual of [32784, 32527, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(2256, 32783, F2, 34) (dual of [32783, 32527, 35]-code), using
- OA 17-folding and stacking [i] based on linear OA(2256, 32776, F2, 34) (dual of [32776, 32520, 35]-code), using
(256−34, 256, 5602)-Net over F2 — Digital
Digital (222, 256, 5602)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2256, 5602, F2, 5, 34) (dual of [(5602, 5), 27754, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2256, 6556, F2, 5, 34) (dual of [(6556, 5), 32524, 35]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2256, 32780, F2, 34) (dual of [32780, 32524, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(2256, 32783, F2, 34) (dual of [32783, 32527, 35]-code), using
- 1 times truncation [i] based on linear OA(2257, 32784, F2, 35) (dual of [32784, 32527, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(32) [i] based on
- linear OA(2256, 32768, F2, 35) (dual of [32768, 32512, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2241, 32768, F2, 33) (dual of [32768, 32527, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [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(34) ⊂ Ce(32) [i] based on
- 1 times truncation [i] based on linear OA(2257, 32784, F2, 35) (dual of [32784, 32527, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(2256, 32783, F2, 34) (dual of [32783, 32527, 35]-code), using
- OOA 5-folding [i] based on linear OA(2256, 32780, F2, 34) (dual of [32780, 32524, 35]-code), using
- discarding factors / shortening the dual code based on linear OOA(2256, 6556, F2, 5, 34) (dual of [(6556, 5), 32524, 35]-NRT-code), using
(256−34, 256, 244939)-Net in Base 2 — Upper bound on s
There is no (222, 256, 244940)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 115797 742704 991502 314060 734837 768907 464106 665644 682422 170759 954861 121403 907468 > 2256 [i]