Best Known (165−17, 165, s)-Nets in Base 2
(165−17, 165, 131075)-Net over F2 — Constructive and digital
Digital (148, 165, 131075)-net over F2, using
- net defined by OOA [i] based on linear OOA(2165, 131075, F2, 17, 17) (dual of [(131075, 17), 2228110, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2165, 1048601, F2, 17) (dual of [1048601, 1048436, 18]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2162, 1048598, F2, 17) (dual of [1048598, 1048436, 18]-code), using
- construction X4 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(221, 22, F2, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,2)), using
- dual of repetition code with length 22 [i]
- linear OA(21, 22, F2, 1) (dual of [22, 21, 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 X4 applied to Ce(16) ⊂ Ce(14) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(2162, 1048598, F2, 17) (dual of [1048598, 1048436, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2165, 1048601, F2, 17) (dual of [1048601, 1048436, 18]-code), using
(165−17, 165, 174766)-Net over F2 — Digital
Digital (148, 165, 174766)-net over F2, using
- 23 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
(165−17, 165, 5582183)-Net in Base 2 — Upper bound on s
There is no (148, 165, 5582184)-net in base 2, because
- 1 times m-reduction [i] would yield (148, 164, 5582184)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 23 384054 525718 158247 917186 625771 799231 705332 865484 > 2164 [i]