Best Known (172, 196, s)-Nets in Base 2
(172, 196, 5462)-Net over F2 — Constructive and digital
Digital (172, 196, 5462)-net over F2, using
- 22 times duplication [i] based on digital (170, 194, 5462)-net over F2, using
- t-expansion [i] based on digital (169, 194, 5462)-net over F2, using
- net defined by OOA [i] based on linear OOA(2194, 5462, F2, 25, 25) (dual of [(5462, 25), 136356, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2194, 65545, F2, 25) (dual of [65545, 65351, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2194, 65553, F2, 25) (dual of [65553, 65359, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- 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(2177, 65536, F2, 23) (dual of [65536, 65359, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2194, 65553, F2, 25) (dual of [65553, 65359, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2194, 65545, F2, 25) (dual of [65545, 65351, 26]-code), using
- net defined by OOA [i] based on linear OOA(2194, 5462, F2, 25, 25) (dual of [(5462, 25), 136356, 26]-NRT-code), using
- t-expansion [i] based on digital (169, 194, 5462)-net over F2, using
(172, 196, 11788)-Net over F2 — Digital
Digital (172, 196, 11788)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2196, 11788, F2, 5, 24) (dual of [(11788, 5), 58744, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2196, 13111, F2, 5, 24) (dual of [(13111, 5), 65359, 25]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2196, 65555, F2, 24) (dual of [65555, 65359, 25]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2193, 65552, F2, 24) (dual of [65552, 65359, 25]-code), using
- 1 times truncation [i] based on linear OA(2194, 65553, F2, 25) (dual of [65553, 65359, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- 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(2177, 65536, F2, 23) (dual of [65536, 65359, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(24) ⊂ Ce(22) [i] based on
- 1 times truncation [i] based on linear OA(2194, 65553, F2, 25) (dual of [65553, 65359, 26]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2193, 65552, F2, 24) (dual of [65552, 65359, 25]-code), using
- OOA 5-folding [i] based on linear OA(2196, 65555, F2, 24) (dual of [65555, 65359, 25]-code), using
- discarding factors / shortening the dual code based on linear OOA(2196, 13111, F2, 5, 24) (dual of [(13111, 5), 65359, 25]-NRT-code), using
(172, 196, 436683)-Net in Base 2 — Upper bound on s
There is no (172, 196, 436684)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 100433 647626 825965 142389 218048 068066 767671 003322 663531 175312 > 2196 [i]