Best Known (107−17, 107, s)-Nets in Base 2
(107−17, 107, 1025)-Net over F2 — Constructive and digital
Digital (90, 107, 1025)-net over F2, using
- 21 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
(107−17, 107, 2012)-Net over F2 — Digital
Digital (90, 107, 2012)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2107, 2012, F2, 4, 17) (dual of [(2012, 4), 7941, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2107, 2051, F2, 4, 17) (dual of [(2051, 4), 8097, 18]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2106, 2051, F2, 4, 17) (dual of [(2051, 4), 8098, 18]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2106, 8204, F2, 17) (dual of [8204, 8098, 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 4-folding [i] based on linear OA(2106, 8204, F2, 17) (dual of [8204, 8098, 18]-code), using
- 21 times duplication [i] based on linear OOA(2106, 2051, F2, 4, 17) (dual of [(2051, 4), 8098, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2107, 2051, F2, 4, 17) (dual of [(2051, 4), 8097, 18]-NRT-code), using
(107−17, 107, 36660)-Net in Base 2 — Upper bound on s
There is no (90, 107, 36661)-net in base 2, because
- 1 times m-reduction [i] would yield (90, 106, 36661)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 81 134107 492297 174453 078531 349222 > 2106 [i]