Best Known (143−15, 143, s)-Nets in Base 2
(143−15, 143, 149799)-Net over F2 — Constructive and digital
Digital (128, 143, 149799)-net over F2, using
- 21 times duplication [i] based on digital (127, 142, 149799)-net over F2, using
- net defined by OOA [i] based on linear OOA(2142, 149799, F2, 15, 15) (dual of [(149799, 15), 2246843, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2142, 1048594, F2, 15) (dual of [1048594, 1048452, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2142, 1048597, F2, 15) (dual of [1048597, 1048455, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(2141, 1048576, F2, 15) (dual of [1048576, 1048435, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2121, 1048576, F2, 13) (dual of [1048576, 1048455, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(14) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(2142, 1048597, F2, 15) (dual of [1048597, 1048455, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2142, 1048594, F2, 15) (dual of [1048594, 1048452, 16]-code), using
- net defined by OOA [i] based on linear OOA(2142, 149799, F2, 15, 15) (dual of [(149799, 15), 2246843, 16]-NRT-code), using
(143−15, 143, 174766)-Net over F2 — Digital
Digital (128, 143, 174766)-net over F2, using
- 21 times duplication [i] based on digital (127, 142, 174766)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2142, 174766, F2, 6, 15) (dual of [(174766, 6), 1048454, 16]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2142, 1048596, F2, 15) (dual of [1048596, 1048454, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2142, 1048597, F2, 15) (dual of [1048597, 1048455, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(2141, 1048576, F2, 15) (dual of [1048576, 1048435, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2121, 1048576, F2, 13) (dual of [1048576, 1048455, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(14) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(2142, 1048597, F2, 15) (dual of [1048597, 1048455, 16]-code), using
- OOA 6-folding [i] based on linear OA(2142, 1048596, F2, 15) (dual of [1048596, 1048454, 16]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2142, 174766, F2, 6, 15) (dual of [(174766, 6), 1048454, 16]-NRT-code), using
(143−15, 143, 4320421)-Net in Base 2 — Upper bound on s
There is no (128, 143, 4320422)-net in base 2, because
- 1 times m-reduction [i] would yield (128, 142, 4320422)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 5 575190 123934 324302 093076 040829 935587 796188 > 2142 [i]