Best Known (201−31, 201, s)-Nets in Base 2
(201−31, 201, 547)-Net over F2 — Constructive and digital
Digital (170, 201, 547)-net over F2, using
- 24 times duplication [i] based on digital (166, 197, 547)-net over F2, using
- net defined by OOA [i] based on linear OOA(2197, 547, F2, 31, 31) (dual of [(547, 31), 16760, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2197, 8206, F2, 31) (dual of [8206, 8009, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(2196, 8192, F2, 31) (dual of [8192, 7996, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2183, 8192, F2, 29) (dual of [8192, 8009, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(30) ⊂ Ce(28) [i] based on
- OOA 15-folding and stacking with additional row [i] based on linear OA(2197, 8206, F2, 31) (dual of [8206, 8009, 32]-code), using
- net defined by OOA [i] based on linear OOA(2197, 547, F2, 31, 31) (dual of [(547, 31), 16760, 32]-NRT-code), using
(201−31, 201, 1977)-Net over F2 — Digital
Digital (170, 201, 1977)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2201, 1977, F2, 4, 31) (dual of [(1977, 4), 7707, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2201, 2052, F2, 4, 31) (dual of [(2052, 4), 8007, 32]-NRT-code), using
- 22 times duplication [i] based on linear OOA(2199, 2052, F2, 4, 31) (dual of [(2052, 4), 8009, 32]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2199, 8208, F2, 31) (dual of [8208, 8009, 32]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2197, 8206, F2, 31) (dual of [8206, 8009, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(2196, 8192, F2, 31) (dual of [8192, 7996, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2183, 8192, F2, 29) (dual of [8192, 8009, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(30) ⊂ Ce(28) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(2197, 8206, F2, 31) (dual of [8206, 8009, 32]-code), using
- OOA 4-folding [i] based on linear OA(2199, 8208, F2, 31) (dual of [8208, 8009, 32]-code), using
- 22 times duplication [i] based on linear OOA(2199, 2052, F2, 4, 31) (dual of [(2052, 4), 8009, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2201, 2052, F2, 4, 31) (dual of [(2052, 4), 8007, 32]-NRT-code), using
(201−31, 201, 66275)-Net in Base 2 — Upper bound on s
There is no (170, 201, 66276)-net in base 2, because
- 1 times m-reduction [i] would yield (170, 200, 66276)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1 606965 206403 884742 765049 828264 847586 948574 210660 965723 265848 > 2200 [i]