Best Known (135−13, 135, s)-Nets in Base 2
(135−13, 135, 699054)-Net over F2 — Constructive and digital
Digital (122, 135, 699054)-net over F2, using
- 21 times duplication [i] based on digital (121, 134, 699054)-net over F2, using
- net defined by OOA [i] based on linear OOA(2134, 699054, F2, 13, 13) (dual of [(699054, 13), 9087568, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2134, 4194325, F2, 13) (dual of [4194325, 4194191, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2134, 4194327, F2, 13) (dual of [4194327, 4194193, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(2133, 4194304, F2, 13) (dual of [4194304, 4194171, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2111, 4194304, F2, 11) (dual of [4194304, 4194193, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(21, 23, F2, 1) (dual of [23, 22, 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(2134, 4194327, F2, 13) (dual of [4194327, 4194193, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2134, 4194325, F2, 13) (dual of [4194325, 4194191, 14]-code), using
- net defined by OOA [i] based on linear OOA(2134, 699054, F2, 13, 13) (dual of [(699054, 13), 9087568, 14]-NRT-code), using
(135−13, 135, 838865)-Net over F2 — Digital
Digital (122, 135, 838865)-net over F2, using
- 21 times duplication [i] based on digital (121, 134, 838865)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2134, 838865, F2, 5, 13) (dual of [(838865, 5), 4194191, 14]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2134, 4194325, F2, 13) (dual of [4194325, 4194191, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2134, 4194327, F2, 13) (dual of [4194327, 4194193, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(2133, 4194304, F2, 13) (dual of [4194304, 4194171, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2111, 4194304, F2, 11) (dual of [4194304, 4194193, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(21, 23, F2, 1) (dual of [23, 22, 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(2134, 4194327, F2, 13) (dual of [4194327, 4194193, 14]-code), using
- OOA 5-folding [i] based on linear OA(2134, 4194325, F2, 13) (dual of [4194325, 4194191, 14]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2134, 838865, F2, 5, 13) (dual of [(838865, 5), 4194191, 14]-NRT-code), using
(135−13, 135, large)-Net in Base 2 — Upper bound on s
There is no (122, 135, large)-net in base 2, because
- 11 times m-reduction [i] would yield (122, 124, large)-net in base 2, but