Best Known (162, 162+27, s)-Nets in Base 2
(162, 162+27, 1262)-Net over F2 — Constructive and digital
Digital (162, 189, 1262)-net over F2, using
- net defined by OOA [i] based on linear OOA(2189, 1262, F2, 27, 27) (dual of [(1262, 27), 33885, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2189, 16407, F2, 27) (dual of [16407, 16218, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2189, 16416, F2, 27) (dual of [16416, 16227, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [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(2155, 16384, F2, 23) (dual of [16384, 16229, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2189, 16416, F2, 27) (dual of [16416, 16227, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2189, 16407, F2, 27) (dual of [16407, 16218, 28]-code), using
(162, 162+27, 3283)-Net over F2 — Digital
Digital (162, 189, 3283)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2189, 3283, F2, 5, 27) (dual of [(3283, 5), 16226, 28]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2189, 16415, F2, 27) (dual of [16415, 16226, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2189, 16416, F2, 27) (dual of [16416, 16227, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [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(2155, 16384, F2, 23) (dual of [16384, 16229, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2189, 16416, F2, 27) (dual of [16416, 16227, 28]-code), using
- OOA 5-folding [i] based on linear OA(2189, 16415, F2, 27) (dual of [16415, 16226, 28]-code), using
(162, 162+27, 127849)-Net in Base 2 — Upper bound on s
There is no (162, 189, 127850)-net in base 2, because
- 1 times m-reduction [i] would yield (162, 188, 127850)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 392 344999 277120 820530 342722 523710 648042 176975 278868 586776 > 2188 [i]