Best Known (223, 253, s)-Nets in Base 2
(223, 253, 4373)-Net over F2 — Constructive and digital
Digital (223, 253, 4373)-net over F2, using
- net defined by OOA [i] based on linear OOA(2253, 4373, F2, 30, 30) (dual of [(4373, 30), 130937, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(2253, 65595, F2, 30) (dual of [65595, 65342, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(2253, 65596, F2, 30) (dual of [65596, 65343, 31]-code), using
- 1 times truncation [i] based on linear OA(2254, 65597, F2, 31) (dual of [65597, 65343, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- linear OA(2241, 65536, F2, 31) (dual of [65536, 65295, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2193, 65536, F2, 25) (dual of [65536, 65343, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(213, 61, F2, 5) (dual of [61, 48, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(213, 63, F2, 5) (dual of [63, 50, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(213, 63, F2, 5) (dual of [63, 50, 6]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- 1 times truncation [i] based on linear OA(2254, 65597, F2, 31) (dual of [65597, 65343, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2253, 65596, F2, 30) (dual of [65596, 65343, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(2253, 65595, F2, 30) (dual of [65595, 65342, 31]-code), using
(223, 253, 12613)-Net over F2 — Digital
Digital (223, 253, 12613)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2253, 12613, F2, 5, 30) (dual of [(12613, 5), 62812, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2253, 13119, F2, 5, 30) (dual of [(13119, 5), 65342, 31]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2253, 65595, F2, 30) (dual of [65595, 65342, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(2253, 65596, F2, 30) (dual of [65596, 65343, 31]-code), using
- 1 times truncation [i] based on linear OA(2254, 65597, F2, 31) (dual of [65597, 65343, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- linear OA(2241, 65536, F2, 31) (dual of [65536, 65295, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2193, 65536, F2, 25) (dual of [65536, 65343, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(213, 61, F2, 5) (dual of [61, 48, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(213, 63, F2, 5) (dual of [63, 50, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(213, 63, F2, 5) (dual of [63, 50, 6]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- 1 times truncation [i] based on linear OA(2254, 65597, F2, 31) (dual of [65597, 65343, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2253, 65596, F2, 30) (dual of [65596, 65343, 31]-code), using
- OOA 5-folding [i] based on linear OA(2253, 65595, F2, 30) (dual of [65595, 65342, 31]-code), using
- discarding factors / shortening the dual code based on linear OOA(2253, 13119, F2, 5, 30) (dual of [(13119, 5), 65342, 31]-NRT-code), using
(223, 253, 767585)-Net in Base 2 — Upper bound on s
There is no (223, 253, 767586)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 14474 140064 438078 100808 689765 610935 230199 899307 222574 265180 380155 031585 990928 > 2253 [i]