Best Known (78−14, 78, s)-Nets in Base 2
(78−14, 78, 294)-Net over F2 — Constructive and digital
Digital (64, 78, 294)-net over F2, using
- net defined by OOA [i] based on linear OOA(278, 294, F2, 14, 14) (dual of [(294, 14), 4038, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(278, 2058, F2, 14) (dual of [2058, 1980, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(278, 2059, F2, 14) (dual of [2059, 1981, 15]-code), using
- 1 times truncation [i] based on linear OA(279, 2060, F2, 15) (dual of [2060, 1981, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(278, 2048, F2, 15) (dual of [2048, 1970, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(267, 2048, F2, 13) (dual of [2048, 1981, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(21, 12, F2, 1) (dual of [12, 11, 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
- 1 times truncation [i] based on linear OA(279, 2060, F2, 15) (dual of [2060, 1981, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(278, 2059, F2, 14) (dual of [2059, 1981, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(278, 2058, F2, 14) (dual of [2058, 1980, 15]-code), using
(78−14, 78, 686)-Net over F2 — Digital
Digital (64, 78, 686)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(278, 686, F2, 3, 14) (dual of [(686, 3), 1980, 15]-NRT-code), using
- OOA 3-folding [i] based on linear OA(278, 2058, F2, 14) (dual of [2058, 1980, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(278, 2059, F2, 14) (dual of [2059, 1981, 15]-code), using
- 1 times truncation [i] based on linear OA(279, 2060, F2, 15) (dual of [2060, 1981, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(278, 2048, F2, 15) (dual of [2048, 1970, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(267, 2048, F2, 13) (dual of [2048, 1981, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(21, 12, F2, 1) (dual of [12, 11, 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
- 1 times truncation [i] based on linear OA(279, 2060, F2, 15) (dual of [2060, 1981, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(278, 2059, F2, 14) (dual of [2059, 1981, 15]-code), using
- OOA 3-folding [i] based on linear OA(278, 2058, F2, 14) (dual of [2058, 1980, 15]-code), using
(78−14, 78, 7632)-Net in Base 2 — Upper bound on s
There is no (64, 78, 7633)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 302283 326099 708686 686464 > 278 [i]