Best Known (106−12, 106, s)-Nets in Base 2
(106−12, 106, 21848)-Net over F2 — Constructive and digital
Digital (94, 106, 21848)-net over F2, using
- 22 times duplication [i] based on digital (92, 104, 21848)-net over F2, using
- t-expansion [i] based on digital (91, 104, 21848)-net over F2, using
- net defined by OOA [i] based on linear OOA(2104, 21848, F2, 13, 13) (dual of [(21848, 13), 283920, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2104, 131089, F2, 13) (dual of [131089, 130985, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2104, 131090, F2, 13) (dual of [131090, 130986, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(2103, 131072, F2, 13) (dual of [131072, 130969, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(286, 131072, F2, 11) (dual of [131072, 130986, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(2104, 131090, F2, 13) (dual of [131090, 130986, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2104, 131089, F2, 13) (dual of [131089, 130985, 14]-code), using
- net defined by OOA [i] based on linear OOA(2104, 21848, F2, 13, 13) (dual of [(21848, 13), 283920, 14]-NRT-code), using
- t-expansion [i] based on digital (91, 104, 21848)-net over F2, using
(106−12, 106, 32773)-Net over F2 — Digital
Digital (94, 106, 32773)-net over F2, using
- 21 times duplication [i] based on digital (93, 105, 32773)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2105, 32773, F2, 4, 12) (dual of [(32773, 4), 130987, 13]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2105, 131092, F2, 12) (dual of [131092, 130987, 13]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2103, 131090, F2, 12) (dual of [131090, 130987, 13]-code), using
- 1 times truncation [i] based on linear OA(2104, 131091, F2, 13) (dual of [131091, 130987, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(2103, 131072, F2, 13) (dual of [131072, 130969, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(286, 131072, F2, 11) (dual of [131072, 130986, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(218, 19, F2, 17) (dual of [19, 1, 18]-code), using
- strength reduction [i] based on linear OA(218, 19, F2, 18) (dual of [19, 1, 19]-code or 19-arc in PG(17,2)), using
- dual of repetition code with length 19 [i]
- strength reduction [i] based on linear OA(218, 19, F2, 18) (dual of [19, 1, 19]-code or 19-arc in PG(17,2)), using
- linear OA(21, 19, F2, 1) (dual of [19, 18, 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 X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- 1 times truncation [i] based on linear OA(2104, 131091, F2, 13) (dual of [131091, 130987, 14]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2103, 131090, F2, 12) (dual of [131090, 130987, 13]-code), using
- OOA 4-folding [i] based on linear OA(2105, 131092, F2, 12) (dual of [131092, 130987, 13]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2105, 32773, F2, 4, 12) (dual of [(32773, 4), 130987, 13]-NRT-code), using
(106−12, 106, 622891)-Net in Base 2 — Upper bound on s
There is no (94, 106, 622892)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 81 129645 860936 543308 466512 531510 > 2106 [i]