Best Known (207−25, 207, s)-Nets in Base 2
(207−25, 207, 10924)-Net over F2 — Constructive and digital
Digital (182, 207, 10924)-net over F2, using
- 21 times duplication [i] based on digital (181, 206, 10924)-net over F2, using
- net defined by OOA [i] based on linear OOA(2206, 10924, F2, 25, 25) (dual of [(10924, 25), 272894, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2206, 131089, F2, 25) (dual of [131089, 130883, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2206, 131090, F2, 25) (dual of [131090, 130884, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(2205, 131072, F2, 25) (dual of [131072, 130867, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2188, 131072, F2, 23) (dual of [131072, 130884, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(21, 18, F2, 1) (dual of [18, 17, 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(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2206, 131090, F2, 25) (dual of [131090, 130884, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2206, 131089, F2, 25) (dual of [131089, 130883, 26]-code), using
- net defined by OOA [i] based on linear OOA(2206, 10924, F2, 25, 25) (dual of [(10924, 25), 272894, 26]-NRT-code), using
(207−25, 207, 18727)-Net over F2 — Digital
Digital (182, 207, 18727)-net over F2, using
- 21 times duplication [i] based on digital (181, 206, 18727)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2206, 18727, F2, 7, 25) (dual of [(18727, 7), 130883, 26]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2206, 131089, F2, 25) (dual of [131089, 130883, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2206, 131090, F2, 25) (dual of [131090, 130884, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(2205, 131072, F2, 25) (dual of [131072, 130867, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2188, 131072, F2, 23) (dual of [131072, 130884, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(21, 18, F2, 1) (dual of [18, 17, 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(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2206, 131090, F2, 25) (dual of [131090, 130884, 26]-code), using
- OOA 7-folding [i] based on linear OA(2206, 131089, F2, 25) (dual of [131089, 130883, 26]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2206, 18727, F2, 7, 25) (dual of [(18727, 7), 130883, 26]-NRT-code), using
(207−25, 207, 778096)-Net in Base 2 — Upper bound on s
There is no (182, 207, 778097)-net in base 2, because
- 1 times m-reduction [i] would yield (182, 206, 778097)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 102 845462 428169 908854 014279 372020 560122 550129 828764 213963 887668 > 2206 [i]