Best Known (222, 222+28, s)-Nets in Base 2
(222, 222+28, 9365)-Net over F2 — Constructive and digital
Digital (222, 250, 9365)-net over F2, using
- 24 times duplication [i] based on digital (218, 246, 9365)-net over F2, using
- t-expansion [i] based on digital (217, 246, 9365)-net over F2, using
- net defined by OOA [i] based on linear OOA(2246, 9365, F2, 29, 29) (dual of [(9365, 29), 271339, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2246, 131111, F2, 29) (dual of [131111, 130865, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2246, 131114, F2, 29) (dual of [131114, 130868, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- linear OA(2239, 131073, F2, 29) (dual of [131073, 130834, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 131073 | 234−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(2205, 131073, F2, 25) (dual of [131073, 130868, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 131073 | 234−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(27, 41, F2, 3) (dual of [41, 34, 4]-code or 41-cap in PG(6,2)), using
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 4 [i]
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2246, 131114, F2, 29) (dual of [131114, 130868, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2246, 131111, F2, 29) (dual of [131111, 130865, 30]-code), using
- net defined by OOA [i] based on linear OOA(2246, 9365, F2, 29, 29) (dual of [(9365, 29), 271339, 30]-NRT-code), using
- t-expansion [i] based on digital (217, 246, 9365)-net over F2, using
(222, 222+28, 21853)-Net over F2 — Digital
Digital (222, 250, 21853)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2250, 21853, F2, 6, 28) (dual of [(21853, 6), 130868, 29]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2250, 131118, F2, 28) (dual of [131118, 130868, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(2250, 131119, F2, 28) (dual of [131119, 130869, 29]-code), using
- 1 times truncation [i] based on linear OA(2251, 131120, F2, 29) (dual of [131120, 130869, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- linear OA(2239, 131072, F2, 29) (dual of [131072, 130833, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(212, 48, F2, 5) (dual of [48, 36, 6]-code), using
- adding a parity check bit [i] based on linear OA(211, 47, F2, 4) (dual of [47, 36, 5]-code), using
- extracting embedded orthogonal array [i] based on digital (7, 11, 47)-net over F2, using
- adding a parity check bit [i] based on linear OA(211, 47, F2, 4) (dual of [47, 36, 5]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- 1 times truncation [i] based on linear OA(2251, 131120, F2, 29) (dual of [131120, 130869, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2250, 131119, F2, 28) (dual of [131119, 130869, 29]-code), using
- OOA 6-folding [i] based on linear OA(2250, 131118, F2, 28) (dual of [131118, 130868, 29]-code), using
(222, 222+28, 1435447)-Net in Base 2 — Upper bound on s
There is no (222, 250, 1435448)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 1809 268332 961854 479893 014291 065116 536863 400606 595464 081863 506852 906619 749714 > 2250 [i]