Best Known (163, 163+26, s)-Nets in Base 2
(163, 163+26, 1262)-Net over F2 — Constructive and digital
Digital (163, 189, 1262)-net over F2, using
- t-expansion [i] based on 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
- net defined by OOA [i] based on linear OOA(2189, 1262, F2, 27, 27) (dual of [(1262, 27), 33885, 28]-NRT-code), using
(163, 163+26, 3864)-Net over F2 — Digital
Digital (163, 189, 3864)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2189, 3864, F2, 4, 26) (dual of [(3864, 4), 15267, 27]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2189, 4104, F2, 4, 26) (dual of [(4104, 4), 16227, 27]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2189, 16416, F2, 26) (dual of [16416, 16227, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(2189, 16418, F2, 26) (dual of [16418, 16229, 27]-code), using
- 1 times truncation [i] based on linear OA(2190, 16419, F2, 27) (dual of [16419, 16229, 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(27, 35, F2, 3) (dual of [35, 28, 4]-code or 35-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(26) ⊂ Ce(22) [i] based on
- 1 times truncation [i] based on linear OA(2190, 16419, F2, 27) (dual of [16419, 16229, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2189, 16418, F2, 26) (dual of [16418, 16229, 27]-code), using
- OOA 4-folding [i] based on linear OA(2189, 16416, F2, 26) (dual of [16416, 16227, 27]-code), using
- discarding factors / shortening the dual code based on linear OOA(2189, 4104, F2, 4, 26) (dual of [(4104, 4), 16227, 27]-NRT-code), using
(163, 163+26, 134852)-Net in Base 2 — Upper bound on s
There is no (163, 189, 134853)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 784 698825 326479 597972 885644 945315 609364 631262 197721 068592 > 2189 [i]