Best Known (109, 123, s)-Nets in Base 2
(109, 123, 18727)-Net over F2 — Constructive and digital
Digital (109, 123, 18727)-net over F2, using
- 22 times duplication [i] based on digital (107, 121, 18727)-net over F2, using
- t-expansion [i] based on digital (106, 121, 18727)-net over F2, using
- net defined by OOA [i] based on linear OOA(2121, 18727, F2, 15, 15) (dual of [(18727, 15), 280784, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2121, 131090, F2, 15) (dual of [131090, 130969, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(2120, 131072, F2, 15) (dual of [131072, 130952, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 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(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(14) ⊂ Ce(12) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(2121, 131090, F2, 15) (dual of [131090, 130969, 16]-code), using
- net defined by OOA [i] based on linear OOA(2121, 18727, F2, 15, 15) (dual of [(18727, 15), 280784, 16]-NRT-code), using
- t-expansion [i] based on digital (106, 121, 18727)-net over F2, using
(109, 123, 32773)-Net over F2 — Digital
Digital (109, 123, 32773)-net over F2, using
- 21 times duplication [i] based on digital (108, 122, 32773)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2122, 32773, F2, 4, 14) (dual of [(32773, 4), 130970, 15]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2122, 131092, F2, 14) (dual of [131092, 130970, 15]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2120, 131090, F2, 14) (dual of [131090, 130970, 15]-code), using
- 1 times truncation [i] based on linear OA(2121, 131091, F2, 15) (dual of [131091, 130970, 16]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(2120, 131072, F2, 15) (dual of [131072, 130952, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 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(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(14) ⊂ Ce(12) [i] based on
- 1 times truncation [i] based on linear OA(2121, 131091, F2, 15) (dual of [131091, 130970, 16]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2120, 131090, F2, 14) (dual of [131090, 130970, 15]-code), using
- OOA 4-folding [i] based on linear OA(2122, 131092, F2, 14) (dual of [131092, 130970, 15]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2122, 32773, F2, 4, 14) (dual of [(32773, 4), 130970, 15]-NRT-code), using
(109, 123, 658323)-Net in Base 2 — Upper bound on s
There is no (109, 123, 658324)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 10 633921 707515 305897 360133 753775 520604 > 2123 [i]