Best Known (67, 67+15, s)-Nets in Base 2
(67, 67+15, 294)-Net over F2 — Constructive and digital
Digital (67, 82, 294)-net over F2, using
- 23 times duplication [i] based on digital (64, 79, 294)-net over F2, using
- net defined by OOA [i] based on linear OOA(279, 294, F2, 15, 15) (dual of [(294, 15), 4331, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(279, 2059, F2, 15) (dual of [2059, 1980, 16]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(279, 2060, F2, 15) (dual of [2060, 1981, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(279, 2059, F2, 15) (dual of [2059, 1980, 16]-code), using
- net defined by OOA [i] based on linear OOA(279, 294, F2, 15, 15) (dual of [(294, 15), 4331, 16]-NRT-code), using
(67, 67+15, 687)-Net over F2 — Digital
Digital (67, 82, 687)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(282, 687, F2, 3, 15) (dual of [(687, 3), 1979, 16]-NRT-code), using
- 22 times duplication [i] based on linear OOA(280, 687, F2, 3, 15) (dual of [(687, 3), 1981, 16]-NRT-code), using
- OOA 3-folding [i] based on linear OA(280, 2061, F2, 15) (dual of [2061, 1981, 16]-code), using
- 1 times code embedding in larger space [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 code embedding in larger space [i] based on linear OA(279, 2060, F2, 15) (dual of [2060, 1981, 16]-code), using
- OOA 3-folding [i] based on linear OA(280, 2061, F2, 15) (dual of [2061, 1981, 16]-code), using
- 22 times duplication [i] based on linear OOA(280, 687, F2, 3, 15) (dual of [(687, 3), 1981, 16]-NRT-code), using
(67, 67+15, 10276)-Net in Base 2 — Upper bound on s
There is no (67, 82, 10277)-net in base 2, because
- 1 times m-reduction [i] would yield (67, 81, 10277)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 2 418741 070012 220722 118616 > 281 [i]