Best Known (80−13, 80, s)-Nets in Base 2
(80−13, 80, 1367)-Net over F2 — Constructive and digital
Digital (67, 80, 1367)-net over F2, using
- net defined by OOA [i] based on linear OOA(280, 1367, F2, 13, 13) (dual of [(1367, 13), 17691, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(280, 8203, F2, 13) (dual of [8203, 8123, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(280, 8206, F2, 13) (dual of [8206, 8126, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(279, 8192, F2, 13) (dual of [8192, 8113, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(266, 8192, F2, 11) (dual of [8192, 8126, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 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(280, 8206, F2, 13) (dual of [8206, 8126, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(280, 8203, F2, 13) (dual of [8203, 8123, 14]-code), using
(80−13, 80, 2051)-Net over F2 — Digital
Digital (67, 80, 2051)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(280, 2051, F2, 4, 13) (dual of [(2051, 4), 8124, 14]-NRT-code), using
- OOA 4-folding [i] based on linear OA(280, 8204, F2, 13) (dual of [8204, 8124, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(280, 8206, F2, 13) (dual of [8206, 8126, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(279, 8192, F2, 13) (dual of [8192, 8113, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(266, 8192, F2, 11) (dual of [8192, 8126, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 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(280, 8206, F2, 13) (dual of [8206, 8126, 14]-code), using
- OOA 4-folding [i] based on linear OA(280, 8204, F2, 13) (dual of [8204, 8124, 14]-code), using
(80−13, 80, 27520)-Net in Base 2 — Upper bound on s
There is no (67, 80, 27521)-net in base 2, because
- 1 times m-reduction [i] would yield (67, 79, 27521)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 604584 957468 107433 910912 > 279 [i]