Best Known (164−23, 164, s)-Nets in Base 2
(164−23, 164, 1492)-Net over F2 — Constructive and digital
Digital (141, 164, 1492)-net over F2, using
- 23 times duplication [i] based on digital (138, 161, 1492)-net over F2, using
- net defined by OOA [i] based on linear OOA(2161, 1492, F2, 23, 23) (dual of [(1492, 23), 34155, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2161, 16413, F2, 23) (dual of [16413, 16252, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2161, 16416, F2, 23) (dual of [16416, 16255, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(2155, 16384, F2, 23) (dual of [16384, 16229, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2127, 16384, F2, 19) (dual of [16384, 16257, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(2161, 16416, F2, 23) (dual of [16416, 16255, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2161, 16413, F2, 23) (dual of [16413, 16252, 24]-code), using
- net defined by OOA [i] based on linear OOA(2161, 1492, F2, 23, 23) (dual of [(1492, 23), 34155, 24]-NRT-code), using
(164−23, 164, 3555)-Net over F2 — Digital
Digital (141, 164, 3555)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2164, 3555, F2, 4, 23) (dual of [(3555, 4), 14056, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2164, 4105, F2, 4, 23) (dual of [(4105, 4), 16256, 24]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2163, 4105, F2, 4, 23) (dual of [(4105, 4), 16257, 24]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2163, 16420, F2, 23) (dual of [16420, 16257, 24]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2162, 16419, F2, 23) (dual of [16419, 16257, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(2155, 16384, F2, 23) (dual of [16384, 16229, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2127, 16384, F2, 19) (dual of [16384, 16257, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(27, 35, F2, 3) (dual of [35, 28, 4]-code or 35-cap in PG(6,2)), using
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 4 [i]
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2162, 16419, F2, 23) (dual of [16419, 16257, 24]-code), using
- OOA 4-folding [i] based on linear OA(2163, 16420, F2, 23) (dual of [16420, 16257, 24]-code), using
- 21 times duplication [i] based on linear OOA(2163, 4105, F2, 4, 23) (dual of [(4105, 4), 16257, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2164, 4105, F2, 4, 23) (dual of [(4105, 4), 16256, 24]-NRT-code), using
(164−23, 164, 141802)-Net in Base 2 — Upper bound on s
There is no (141, 164, 141803)-net in base 2, because
- 1 times m-reduction [i] would yield (141, 163, 141803)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 11 692916 832874 352701 147201 447015 507740 430583 884526 > 2163 [i]