Best Known (172−27, 172, s)-Nets in Base 2
(172−27, 172, 631)-Net over F2 — Constructive and digital
Digital (145, 172, 631)-net over F2, using
- 21 times duplication [i] based on digital (144, 171, 631)-net over F2, using
- net defined by OOA [i] based on linear OOA(2171, 631, F2, 27, 27) (dual of [(631, 27), 16866, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2171, 8204, F2, 27) (dual of [8204, 8033, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2171, 8206, F2, 27) (dual of [8206, 8035, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- linear OA(2170, 8192, F2, 27) (dual of [8192, 8022, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2157, 8192, F2, 25) (dual of [8192, 8035, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 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(2171, 8206, F2, 27) (dual of [8206, 8035, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2171, 8204, F2, 27) (dual of [8204, 8033, 28]-code), using
- net defined by OOA [i] based on linear OOA(2171, 631, F2, 27, 27) (dual of [(631, 27), 16866, 28]-NRT-code), using
(172−27, 172, 1770)-Net over F2 — Digital
Digital (145, 172, 1770)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2172, 1770, F2, 4, 27) (dual of [(1770, 4), 6908, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2172, 2051, F2, 4, 27) (dual of [(2051, 4), 8032, 28]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2171, 2051, F2, 4, 27) (dual of [(2051, 4), 8033, 28]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2171, 8204, F2, 27) (dual of [8204, 8033, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2171, 8206, F2, 27) (dual of [8206, 8035, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- linear OA(2170, 8192, F2, 27) (dual of [8192, 8022, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2157, 8192, F2, 25) (dual of [8192, 8035, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 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(2171, 8206, F2, 27) (dual of [8206, 8035, 28]-code), using
- OOA 4-folding [i] based on linear OA(2171, 8204, F2, 27) (dual of [8204, 8033, 28]-code), using
- 21 times duplication [i] based on linear OOA(2171, 2051, F2, 4, 27) (dual of [(2051, 4), 8033, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2172, 2051, F2, 4, 27) (dual of [(2051, 4), 8032, 28]-NRT-code), using
(172−27, 172, 51635)-Net in Base 2 — Upper bound on s
There is no (145, 172, 51636)-net in base 2, because
- 1 times m-reduction [i] would yield (145, 171, 51636)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 2993 505740 006016 908102 869246 683660 569095 475145 309804 > 2171 [i]