Best Known (241−24, 241, s)-Nets in Base 2
(241−24, 241, 87383)-Net over F2 — Constructive and digital
Digital (217, 241, 87383)-net over F2, using
- net defined by OOA [i] based on linear OOA(2241, 87383, F2, 24, 24) (dual of [(87383, 24), 2096951, 25]-NRT-code), using
- OA 12-folding and stacking [i] 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
- OA 12-folding and stacking [i] based on linear OA(2241, 1048596, F2, 24) (dual of [1048596, 1048355, 25]-code), using
(241−24, 241, 149799)-Net over F2 — Digital
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
(241−24, 241, 5875514)-Net in Base 2 — Upper bound on s
There is no (217, 241, 5875515)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 3 533701 251946 877205 253452 274595 005483 553813 380393 599454 717649 532227 452548 > 2241 [i]