Best Known (133, 133+22, s)-Nets in Base 2
(133, 133+22, 1490)-Net over F2 — Constructive and digital
Digital (133, 155, 1490)-net over F2, using
- net defined by OOA [i] based on linear OOA(2155, 1490, F2, 22, 22) (dual of [(1490, 22), 32625, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(2155, 16390, F2, 22) (dual of [16390, 16235, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(2155, 16398, F2, 22) (dual of [16398, 16243, 23]-code), using
- 1 times truncation [i] based on linear OA(2156, 16399, F2, 23) (dual of [16399, 16243, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- 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(2141, 16384, F2, 21) (dual of [16384, 16243, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(21, 15, F2, 1) (dual of [15, 14, 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(22) ⊂ Ce(20) [i] based on
- 1 times truncation [i] based on linear OA(2156, 16399, F2, 23) (dual of [16399, 16243, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2155, 16398, F2, 22) (dual of [16398, 16243, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(2155, 16390, F2, 22) (dual of [16390, 16235, 23]-code), using
(133, 133+22, 3362)-Net over F2 — Digital
Digital (133, 155, 3362)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2155, 3362, F2, 4, 22) (dual of [(3362, 4), 13293, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2155, 4099, F2, 4, 22) (dual of [(4099, 4), 16241, 23]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2155, 16396, F2, 22) (dual of [16396, 16241, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(2155, 16398, F2, 22) (dual of [16398, 16243, 23]-code), using
- 1 times truncation [i] based on linear OA(2156, 16399, F2, 23) (dual of [16399, 16243, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- 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(2141, 16384, F2, 21) (dual of [16384, 16243, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(21, 15, F2, 1) (dual of [15, 14, 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(22) ⊂ Ce(20) [i] based on
- 1 times truncation [i] based on linear OA(2156, 16399, F2, 23) (dual of [16399, 16243, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2155, 16398, F2, 22) (dual of [16398, 16243, 23]-code), using
- OOA 4-folding [i] based on linear OA(2155, 16396, F2, 22) (dual of [16396, 16241, 23]-code), using
- discarding factors / shortening the dual code based on linear OOA(2155, 4099, F2, 4, 22) (dual of [(4099, 4), 16241, 23]-NRT-code), using
(133, 133+22, 85648)-Net in Base 2 — Upper bound on s
There is no (133, 155, 85649)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 45675 343121 314864 848179 537853 404165 383646 387440 > 2155 [i]