Best Known (187, 187+27, s)-Nets in Base 2
(187, 187+27, 5042)-Net over F2 — Constructive and digital
Digital (187, 214, 5042)-net over F2, using
- 24 times duplication [i] based on digital (183, 210, 5042)-net over F2, using
- net defined by OOA [i] based on linear OOA(2210, 5042, F2, 27, 27) (dual of [(5042, 27), 135924, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2210, 65547, F2, 27) (dual of [65547, 65337, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2210, 65553, F2, 27) (dual of [65553, 65343, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- linear OA(2209, 65536, F2, 27) (dual of [65536, 65327, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2193, 65536, F2, 25) (dual of [65536, 65343, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(21, 17, F2, 1) (dual of [17, 16, 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(2210, 65553, F2, 27) (dual of [65553, 65343, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2210, 65547, F2, 27) (dual of [65547, 65337, 28]-code), using
- net defined by OOA [i] based on linear OOA(2210, 5042, F2, 27, 27) (dual of [(5042, 27), 135924, 28]-NRT-code), using
(187, 187+27, 10926)-Net over F2 — Digital
Digital (187, 214, 10926)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2214, 10926, F2, 6, 27) (dual of [(10926, 6), 65342, 28]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2213, 10926, F2, 6, 27) (dual of [(10926, 6), 65343, 28]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2213, 65556, F2, 27) (dual of [65556, 65343, 28]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2210, 65553, F2, 27) (dual of [65553, 65343, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- linear OA(2209, 65536, F2, 27) (dual of [65536, 65327, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2193, 65536, F2, 25) (dual of [65536, 65343, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(21, 17, F2, 1) (dual of [17, 16, 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
- 3 times code embedding in larger space [i] based on linear OA(2210, 65553, F2, 27) (dual of [65553, 65343, 28]-code), using
- OOA 6-folding [i] based on linear OA(2213, 65556, F2, 27) (dual of [65556, 65343, 28]-code), using
- 21 times duplication [i] based on linear OOA(2213, 10926, F2, 6, 27) (dual of [(10926, 6), 65343, 28]-NRT-code), using
(187, 187+27, 484897)-Net in Base 2 — Upper bound on s
There is no (187, 214, 484898)-net in base 2, because
- 1 times m-reduction [i] would yield (187, 213, 484898)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 13164 246491 100151 549851 066381 478027 644807 524078 158948 886311 247928 > 2213 [i]