Best Known (123−13, 123, s)-Nets in Base 2
(123−13, 123, 174766)-Net over F2 — Constructive and digital
Digital (110, 123, 174766)-net over F2, using
- 21 times duplication [i] based on digital (109, 122, 174766)-net over F2, using
- net defined by OOA [i] based on linear OOA(2122, 174766, F2, 13, 13) (dual of [(174766, 13), 2271836, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2122, 1048597, F2, 13) (dual of [1048597, 1048475, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- 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(2101, 1048576, F2, 11) (dual of [1048576, 1048475, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(12) ⊂ Ce(10) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(2122, 1048597, F2, 13) (dual of [1048597, 1048475, 14]-code), using
- net defined by OOA [i] based on linear OOA(2122, 174766, F2, 13, 13) (dual of [(174766, 13), 2271836, 14]-NRT-code), using
(123−13, 123, 209719)-Net over F2 — Digital
Digital (110, 123, 209719)-net over F2, using
- 21 times duplication [i] based on digital (109, 122, 209719)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2122, 209719, F2, 5, 13) (dual of [(209719, 5), 1048473, 14]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2122, 1048595, F2, 13) (dual of [1048595, 1048473, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2122, 1048597, F2, 13) (dual of [1048597, 1048475, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- 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(2101, 1048576, F2, 11) (dual of [1048576, 1048475, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(2122, 1048597, F2, 13) (dual of [1048597, 1048475, 14]-code), using
- OOA 5-folding [i] based on linear OA(2122, 1048595, F2, 13) (dual of [1048595, 1048473, 14]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2122, 209719, F2, 5, 13) (dual of [(209719, 5), 1048473, 14]-NRT-code), using
(123−13, 123, 3955163)-Net in Base 2 — Upper bound on s
There is no (110, 123, 3955164)-net in base 2, because
- 1 times m-reduction [i] would yield (110, 122, 3955164)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 5 316919 443406 141806 194321 989248 931850 > 2122 [i]