Best Known (220, 220+24, s)-Nets in Base 2
(220, 220+24, 87383)-Net over F2 — Constructive and digital
Digital (220, 244, 87383)-net over F2, using
- 22 times duplication [i] based on digital (218, 242, 87383)-net over F2, using
- t-expansion [i] based on digital (217, 242, 87383)-net over F2, using
- net defined by OOA [i] based on linear OOA(2242, 87383, F2, 25, 25) (dual of [(87383, 25), 2184333, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2242, 1048597, F2, 25) (dual of [1048597, 1048355, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(2241, 1048576, F2, 25) (dual of [1048576, 1048335, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2221, 1048576, F2, 23) (dual of [1048576, 1048355, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(21, 21, F2, 1) (dual of [21, 20, 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
- OOA 12-folding and stacking with additional row [i] based on linear OA(2242, 1048597, F2, 25) (dual of [1048597, 1048355, 26]-code), using
- net defined by OOA [i] based on linear OOA(2242, 87383, F2, 25, 25) (dual of [(87383, 25), 2184333, 26]-NRT-code), using
- t-expansion [i] based on digital (217, 242, 87383)-net over F2, using
(220, 220+24, 149799)-Net over F2 — Digital
Digital (220, 244, 149799)-net over F2, using
- 23 times duplication [i] based on digital (217, 241, 149799)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2241, 149799, F2, 7, 24) (dual of [(149799, 7), 1048352, 25]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2241, 1048593, F2, 24) (dual of [1048593, 1048352, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(2241, 1048596, F2, 24) (dual of [1048596, 1048355, 25]-code), using
- 1 times truncation [i] based on linear OA(2242, 1048597, F2, 25) (dual of [1048597, 1048355, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(2241, 1048576, F2, 25) (dual of [1048576, 1048335, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2221, 1048576, F2, 23) (dual of [1048576, 1048355, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(21, 21, F2, 1) (dual of [21, 20, 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
- 1 times truncation [i] based on linear OA(2242, 1048597, F2, 25) (dual of [1048597, 1048355, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2241, 1048596, F2, 24) (dual of [1048596, 1048355, 25]-code), using
- OOA 7-folding [i] based on linear OA(2241, 1048593, F2, 24) (dual of [1048593, 1048352, 25]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2241, 149799, F2, 7, 24) (dual of [(149799, 7), 1048352, 25]-NRT-code), using
(220, 220+24, 6987206)-Net in Base 2 — Upper bound on s
There is no (220, 244, 6987207)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 28 269583 146531 523961 590715 894638 390892 204539 804944 721983 194109 005951 386344 > 2244 [i]