Best Known (108−17, 108, s)-Nets in Base 2
(108−17, 108, 1025)-Net over F2 — Constructive and digital
Digital (91, 108, 1025)-net over F2, using
- 22 times duplication [i] based on digital (89, 106, 1025)-net over F2, using
- net defined by OOA [i] based on linear OOA(2106, 1025, F2, 17, 17) (dual of [(1025, 17), 17319, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2106, 8201, F2, 17) (dual of [8201, 8095, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2106, 8206, F2, 17) (dual of [8206, 8100, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(2105, 8192, F2, 17) (dual of [8192, 8087, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(292, 8192, F2, 15) (dual of [8192, 8100, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 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(16) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(2106, 8206, F2, 17) (dual of [8206, 8100, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2106, 8201, F2, 17) (dual of [8201, 8095, 18]-code), using
- net defined by OOA [i] based on linear OOA(2106, 1025, F2, 17, 17) (dual of [(1025, 17), 17319, 18]-NRT-code), using
(108−17, 108, 2052)-Net over F2 — Digital
Digital (91, 108, 2052)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2108, 2052, F2, 4, 17) (dual of [(2052, 4), 8100, 18]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2108, 8208, F2, 17) (dual of [8208, 8100, 18]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2106, 8206, F2, 17) (dual of [8206, 8100, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(2105, 8192, F2, 17) (dual of [8192, 8087, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(292, 8192, F2, 15) (dual of [8192, 8100, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 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(16) ⊂ Ce(14) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(2106, 8206, F2, 17) (dual of [8206, 8100, 18]-code), using
- OOA 4-folding [i] based on linear OA(2108, 8208, F2, 17) (dual of [8208, 8100, 18]-code), using
(108−17, 108, 39979)-Net in Base 2 — Upper bound on s
There is no (91, 108, 39980)-net in base 2, because
- 1 times m-reduction [i] would yield (91, 107, 39980)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 162 263522 926249 354422 354753 166337 > 2107 [i]