Best Known (163, 163+18, s)-Nets in Base 2
(163, 163+18, 116510)-Net over F2 — Constructive and digital
Digital (163, 181, 116510)-net over F2, using
- net defined by OOA [i] based on linear OOA(2181, 116510, F2, 18, 18) (dual of [(116510, 18), 2096999, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(2181, 1048590, F2, 18) (dual of [1048590, 1048409, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2181, 1048597, F2, 18) (dual of [1048597, 1048416, 19]-code), using
- 1 times truncation [i] based on linear OA(2182, 1048598, F2, 19) (dual of [1048598, 1048416, 20]-code), using
- construction X4 applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(2181, 1048576, F2, 19) (dual of [1048576, 1048395, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 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(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(18) ⊂ Ce(16) [i] based on
- 1 times truncation [i] based on linear OA(2182, 1048598, F2, 19) (dual of [1048598, 1048416, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2181, 1048597, F2, 18) (dual of [1048597, 1048416, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(2181, 1048590, F2, 18) (dual of [1048590, 1048409, 19]-code), using
(163, 163+18, 174766)-Net over F2 — Digital
Digital (163, 181, 174766)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2181, 174766, F2, 6, 18) (dual of [(174766, 6), 1048415, 19]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2181, 1048596, F2, 18) (dual of [1048596, 1048415, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2181, 1048597, F2, 18) (dual of [1048597, 1048416, 19]-code), using
- 1 times truncation [i] based on linear OA(2182, 1048598, F2, 19) (dual of [1048598, 1048416, 20]-code), using
- construction X4 applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(2181, 1048576, F2, 19) (dual of [1048576, 1048395, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 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(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(18) ⊂ Ce(16) [i] based on
- 1 times truncation [i] based on linear OA(2182, 1048598, F2, 19) (dual of [1048598, 1048416, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2181, 1048597, F2, 18) (dual of [1048597, 1048416, 19]-code), using
- OOA 6-folding [i] based on linear OA(2181, 1048596, F2, 18) (dual of [1048596, 1048415, 19]-code), using
(163, 163+18, 4696755)-Net in Base 2 — Upper bound on s
There is no (163, 181, 4696756)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 3 064995 034912 428663 286403 234159 452011 001391 290204 752898 > 2181 [i]