Best Known (105−13, 105, s)-Nets in Base 2
(105−13, 105, 21848)-Net over F2 — Constructive and digital
Digital (92, 105, 21848)-net over F2, using
- 21 times duplication [i] based on digital (91, 104, 21848)-net over F2, using
- net defined by OOA [i] based on linear OOA(2104, 21848, F2, 13, 13) (dual of [(21848, 13), 283920, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2104, 131089, F2, 13) (dual of [131089, 130985, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2104, 131090, F2, 13) (dual of [131090, 130986, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(2103, 131072, F2, 13) (dual of [131072, 130969, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(286, 131072, F2, 11) (dual of [131072, 130986, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(21, 18, F2, 1) (dual of [18, 17, 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(2104, 131090, F2, 13) (dual of [131090, 130986, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2104, 131089, F2, 13) (dual of [131089, 130985, 14]-code), using
- net defined by OOA [i] based on linear OOA(2104, 21848, F2, 13, 13) (dual of [(21848, 13), 283920, 14]-NRT-code), using
(105−13, 105, 26218)-Net over F2 — Digital
Digital (92, 105, 26218)-net over F2, using
- 21 times duplication [i] based on digital (91, 104, 26218)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2104, 26218, F2, 5, 13) (dual of [(26218, 5), 130986, 14]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2104, 131090, F2, 13) (dual of [131090, 130986, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(2103, 131072, F2, 13) (dual of [131072, 130969, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(286, 131072, F2, 11) (dual of [131072, 130986, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(21, 18, F2, 1) (dual of [18, 17, 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 5-folding [i] based on linear OA(2104, 131090, F2, 13) (dual of [131090, 130986, 14]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2104, 26218, F2, 5, 13) (dual of [(26218, 5), 130986, 14]-NRT-code), using
(105−13, 105, 494387)-Net in Base 2 — Upper bound on s
There is no (92, 105, 494388)-net in base 2, because
- 1 times m-reduction [i] would yield (92, 104, 494388)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 20 282422 662659 418775 344667 024036 > 2104 [i]