Best Known (224−23, 224, s)-Nets in Base 2
(224−23, 224, 95327)-Net over F2 — Constructive and digital
Digital (201, 224, 95327)-net over F2, using
- 21 times duplication [i] based on digital (200, 223, 95327)-net over F2, using
- net defined by OOA [i] based on linear OOA(2223, 95327, F2, 23, 23) (dual of [(95327, 23), 2192298, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2223, 1048598, F2, 23) (dual of [1048598, 1048375, 24]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2222, 1048597, F2, 23) (dual of [1048597, 1048375, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- 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(2201, 1048576, F2, 21) (dual of [1048576, 1048375, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(22) ⊂ Ce(20) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2222, 1048597, F2, 23) (dual of [1048597, 1048375, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2223, 1048598, F2, 23) (dual of [1048598, 1048375, 24]-code), using
- net defined by OOA [i] based on linear OOA(2223, 95327, F2, 23, 23) (dual of [(95327, 23), 2192298, 24]-NRT-code), using
(224−23, 224, 145413)-Net over F2 — Digital
Digital (201, 224, 145413)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2224, 145413, F2, 7, 23) (dual of [(145413, 7), 1017667, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2224, 149799, F2, 7, 23) (dual of [(149799, 7), 1048369, 24]-NRT-code), using
- 22 times duplication [i] based on linear OOA(2222, 149799, F2, 7, 23) (dual of [(149799, 7), 1048371, 24]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2222, 1048593, F2, 23) (dual of [1048593, 1048371, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2222, 1048597, F2, 23) (dual of [1048597, 1048375, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- 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(2201, 1048576, F2, 21) (dual of [1048576, 1048375, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(22) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(2222, 1048597, F2, 23) (dual of [1048597, 1048375, 24]-code), using
- OOA 7-folding [i] based on linear OA(2222, 1048593, F2, 23) (dual of [1048593, 1048371, 24]-code), using
- 22 times duplication [i] based on linear OOA(2222, 149799, F2, 7, 23) (dual of [(149799, 7), 1048371, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2224, 149799, F2, 7, 23) (dual of [(149799, 7), 1048369, 24]-NRT-code), using
(224−23, 224, 6218878)-Net in Base 2 — Upper bound on s
There is no (201, 224, 6218879)-net in base 2, because
- 1 times m-reduction [i] would yield (201, 223, 6218879)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 13 479995 540865 654696 566906 065688 279151 040592 582145 726978 931600 132310 > 2223 [i]