Best Known (111, 111+19, s)-Nets in Base 2
(111, 111+19, 1822)-Net over F2 — Constructive and digital
Digital (111, 130, 1822)-net over F2, using
- 22 times duplication [i] based on digital (109, 128, 1822)-net over F2, using
- net defined by OOA [i] based on linear OOA(2128, 1822, F2, 19, 19) (dual of [(1822, 19), 34490, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2128, 16399, F2, 19) (dual of [16399, 16271, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(2127, 16384, F2, 19) (dual of [16384, 16257, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2113, 16384, F2, 17) (dual of [16384, 16271, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(18) ⊂ Ce(16) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(2128, 16399, F2, 19) (dual of [16399, 16271, 20]-code), using
- net defined by OOA [i] based on linear OOA(2128, 1822, F2, 19, 19) (dual of [(1822, 19), 34490, 20]-NRT-code), using
(111, 111+19, 3280)-Net over F2 — Digital
Digital (111, 130, 3280)-net over F2, using
- 21 times duplication [i] based on digital (110, 129, 3280)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2129, 3280, F2, 5, 19) (dual of [(3280, 5), 16271, 20]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2129, 16400, F2, 19) (dual of [16400, 16271, 20]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2128, 16399, F2, 19) (dual of [16399, 16271, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(2127, 16384, F2, 19) (dual of [16384, 16257, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2113, 16384, F2, 17) (dual of [16384, 16271, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(18) ⊂ Ce(16) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2128, 16399, F2, 19) (dual of [16399, 16271, 20]-code), using
- OOA 5-folding [i] based on linear OA(2129, 16400, F2, 19) (dual of [16400, 16271, 20]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2129, 3280, F2, 5, 19) (dual of [(3280, 5), 16271, 20]-NRT-code), using
(111, 111+19, 85595)-Net in Base 2 — Upper bound on s
There is no (111, 130, 85596)-net in base 2, because
- 1 times m-reduction [i] would yield (111, 129, 85596)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 680 631028 357436 964206 961557 593330 833191 > 2129 [i]