Best Known (219, 219+36, s)-Nets in Base 2
(219, 219+36, 911)-Net over F2 — Constructive and digital
Digital (219, 255, 911)-net over F2, using
- 21 times duplication [i] based on digital (218, 254, 911)-net over F2, using
- t-expansion [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
- t-expansion [i] based on digital (217, 254, 911)-net over F2, using
(219, 219+36, 3354)-Net over F2 — Digital
Digital (219, 255, 3354)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2255, 3354, F2, 4, 36) (dual of [(3354, 4), 13161, 37]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2255, 4100, F2, 4, 36) (dual of [(4100, 4), 16145, 37]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2255, 16400, F2, 36) (dual of [16400, 16145, 37]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2253, 16398, F2, 36) (dual of [16398, 16145, 37]-code), using
- 1 times truncation [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 truncation [i] based on linear OA(2254, 16399, F2, 37) (dual of [16399, 16145, 38]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2253, 16398, F2, 36) (dual of [16398, 16145, 37]-code), using
- OOA 4-folding [i] based on linear OA(2255, 16400, F2, 36) (dual of [16400, 16145, 37]-code), using
- discarding factors / shortening the dual code based on linear OOA(2255, 4100, F2, 4, 36) (dual of [(4100, 4), 16145, 37]-NRT-code), using
(219, 219+36, 138879)-Net in Base 2 — Upper bound on s
There is no (219, 255, 138880)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 57898 203108 267150 604856 071249 605159 528445 628860 784579 828770 955484 510855 117193 > 2255 [i]