Best Known (164−17, 164, s)-Nets in Base 2
(164−17, 164, 131074)-Net over F2 — Constructive and digital
Digital (147, 164, 131074)-net over F2, using
- 22 times duplication [i] based on digital (145, 162, 131074)-net over F2, using
- net defined by OOA [i] based on linear OOA(2162, 131074, F2, 17, 17) (dual of [(131074, 17), 2228096, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2162, 1048593, F2, 17) (dual of [1048593, 1048431, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2162, 1048597, F2, 17) (dual of [1048597, 1048435, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(2161, 1048576, F2, 17) (dual of [1048576, 1048415, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2141, 1048576, F2, 15) (dual of [1048576, 1048435, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(21, 21, F2, 1) (dual of [21, 20, 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(16) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(2162, 1048597, F2, 17) (dual of [1048597, 1048435, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2162, 1048593, F2, 17) (dual of [1048593, 1048431, 18]-code), using
- net defined by OOA [i] based on linear OOA(2162, 131074, F2, 17, 17) (dual of [(131074, 17), 2228096, 18]-NRT-code), using
(164−17, 164, 174766)-Net over F2 — Digital
Digital (147, 164, 174766)-net over F2, using
- 22 times duplication [i] based on digital (145, 162, 174766)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2162, 174766, F2, 6, 17) (dual of [(174766, 6), 1048434, 18]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2162, 1048596, F2, 17) (dual of [1048596, 1048434, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2162, 1048597, F2, 17) (dual of [1048597, 1048435, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(2161, 1048576, F2, 17) (dual of [1048576, 1048415, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2141, 1048576, F2, 15) (dual of [1048576, 1048435, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(21, 21, F2, 1) (dual of [21, 20, 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(16) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(2162, 1048597, F2, 17) (dual of [1048597, 1048435, 18]-code), using
- OOA 6-folding [i] based on linear OA(2162, 1048596, F2, 17) (dual of [1048596, 1048434, 18]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2162, 174766, F2, 6, 17) (dual of [(174766, 6), 1048434, 18]-NRT-code), using
(164−17, 164, 5118883)-Net in Base 2 — Upper bound on s
There is no (147, 164, 5118884)-net in base 2, because
- 1 times m-reduction [i] would yield (147, 163, 5118884)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 11 692020 986894 017392 097340 118594 413257 659637 403804 > 2163 [i]