Best Known (257−37, 257, s)-Nets in Base 2
(257−37, 257, 911)-Net over F2 — Constructive and digital
Digital (220, 257, 911)-net over F2, using
- 23 times duplication [i] based on digital (217, 254, 911)-net over F2, using
- net defined by OOA [i] based on linear OOA(2254, 911, F2, 37, 37) (dual of [(911, 37), 33453, 38]-NRT-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(2254, 16399, F2, 37) (dual of [16399, 16145, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- linear OA(2253, 16384, F2, 37) (dual of [16384, 16131, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(2239, 16384, F2, 35) (dual of [16384, 16145, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [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(36) ⊂ Ce(34) [i] based on
- OOA 18-folding and stacking with additional row [i] based on linear OA(2254, 16399, F2, 37) (dual of [16399, 16145, 38]-code), using
- net defined by OOA [i] based on linear OOA(2254, 911, F2, 37, 37) (dual of [(911, 37), 33453, 38]-NRT-code), using
(257−37, 257, 3280)-Net over F2 — Digital
Digital (220, 257, 3280)-net over F2, using
- 21 times duplication [i] based on digital (219, 256, 3280)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2256, 3280, F2, 5, 37) (dual of [(3280, 5), 16144, 38]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2255, 3280, F2, 5, 37) (dual of [(3280, 5), 16145, 38]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2255, 16400, F2, 37) (dual of [16400, 16145, 38]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2254, 16399, F2, 37) (dual of [16399, 16145, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- linear OA(2253, 16384, F2, 37) (dual of [16384, 16131, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(2239, 16384, F2, 35) (dual of [16384, 16145, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [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(36) ⊂ Ce(34) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2254, 16399, F2, 37) (dual of [16399, 16145, 38]-code), using
- OOA 5-folding [i] based on linear OA(2255, 16400, F2, 37) (dual of [16400, 16145, 38]-code), using
- 21 times duplication [i] based on linear OOA(2255, 3280, F2, 5, 37) (dual of [(3280, 5), 16145, 38]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2256, 3280, F2, 5, 37) (dual of [(3280, 5), 16144, 38]-NRT-code), using
(257−37, 257, 144333)-Net in Base 2 — Upper bound on s
There is no (220, 257, 144334)-net in base 2, because
- 1 times m-reduction [i] would yield (220, 256, 144334)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 115805 625472 779382 080862 036660 217047 812648 514889 936701 730159 128621 758727 819720 > 2256 [i]