Best Known (172, 172+23, s)-Nets in Base 2
(172, 172+23, 11919)-Net over F2 — Constructive and digital
Digital (172, 195, 11919)-net over F2, using
- net defined by OOA [i] based on linear OOA(2195, 11919, F2, 23, 23) (dual of [(11919, 23), 273942, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2195, 131110, F2, 23) (dual of [131110, 130915, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2195, 131113, F2, 23) (dual of [131113, 130918, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(2188, 131072, F2, 23) (dual of [131072, 130884, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2154, 131072, F2, 19) (dual of [131072, 130918, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(27, 41, F2, 3) (dual of [41, 34, 4]-code or 41-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
- discarding factors / shortening the dual code based on linear OA(2195, 131113, F2, 23) (dual of [131113, 130918, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2195, 131110, F2, 23) (dual of [131110, 130915, 24]-code), using
(172, 172+23, 21852)-Net over F2 — Digital
Digital (172, 195, 21852)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2195, 21852, F2, 6, 23) (dual of [(21852, 6), 130917, 24]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2195, 131112, F2, 23) (dual of [131112, 130917, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2195, 131113, F2, 23) (dual of [131113, 130918, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(2188, 131072, F2, 23) (dual of [131072, 130884, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2154, 131072, F2, 19) (dual of [131072, 130918, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(27, 41, F2, 3) (dual of [41, 34, 4]-code or 41-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
- discarding factors / shortening the dual code based on linear OA(2195, 131113, F2, 23) (dual of [131113, 130918, 24]-code), using
- OOA 6-folding [i] based on linear OA(2195, 131112, F2, 23) (dual of [131112, 130917, 24]-code), using
(172, 172+23, 1000188)-Net in Base 2 — Upper bound on s
There is no (172, 195, 1000189)-net in base 2, because
- 1 times m-reduction [i] would yield (172, 194, 1000189)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 25108 658798 544815 067242 440923 246680 817751 499335 885237 138400 > 2194 [i]