Best Known (186−27, 186, s)-Nets in Base 2
(186−27, 186, 1261)-Net over F2 — Constructive and digital
Digital (159, 186, 1261)-net over F2, using
- 22 times duplication [i] based on digital (157, 184, 1261)-net over F2, using
- net defined by OOA [i] based on linear OOA(2184, 1261, F2, 27, 27) (dual of [(1261, 27), 33863, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2184, 16394, F2, 27) (dual of [16394, 16210, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2184, 16399, F2, 27) (dual of [16399, 16215, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- linear OA(2183, 16384, F2, 27) (dual of [16384, 16201, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2169, 16384, F2, 25) (dual of [16384, 16215, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(21, 15, F2, 1) (dual of [15, 14, 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(26) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(2184, 16399, F2, 27) (dual of [16399, 16215, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2184, 16394, F2, 27) (dual of [16394, 16210, 28]-code), using
- net defined by OOA [i] based on linear OOA(2184, 1261, F2, 27, 27) (dual of [(1261, 27), 33863, 28]-NRT-code), using
(186−27, 186, 3280)-Net over F2 — Digital
Digital (159, 186, 3280)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2186, 3280, F2, 5, 27) (dual of [(3280, 5), 16214, 28]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2185, 3280, F2, 5, 27) (dual of [(3280, 5), 16215, 28]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2185, 16400, F2, 27) (dual of [16400, 16215, 28]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2184, 16399, F2, 27) (dual of [16399, 16215, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- linear OA(2183, 16384, F2, 27) (dual of [16384, 16201, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2169, 16384, F2, 25) (dual of [16384, 16215, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(21, 15, F2, 1) (dual of [15, 14, 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(26) ⊂ Ce(24) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2184, 16399, F2, 27) (dual of [16399, 16215, 28]-code), using
- OOA 5-folding [i] based on linear OA(2185, 16400, F2, 27) (dual of [16400, 16215, 28]-code), using
- 21 times duplication [i] based on linear OOA(2185, 3280, F2, 5, 27) (dual of [(3280, 5), 16215, 28]-NRT-code), using
(186−27, 186, 108947)-Net in Base 2 — Upper bound on s
There is no (159, 186, 108948)-net in base 2, because
- 1 times m-reduction [i] would yield (159, 185, 108948)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 49 040459 759708 905001 695178 174669 880851 899207 326276 207028 > 2185 [i]