Best Known (65, 78, s)-Nets in Base 2
(65, 78, 685)-Net over F2 — Constructive and digital
Digital (65, 78, 685)-net over F2, using
- 23 times duplication [i] based on digital (62, 75, 685)-net over F2, using
- net defined by OOA [i] based on linear OOA(275, 685, F2, 13, 13) (dual of [(685, 13), 8830, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(275, 4111, F2, 13) (dual of [4111, 4036, 14]-code), using
- 1 times code embedding in larger space [i] based on linear OA(274, 4110, F2, 13) (dual of [4110, 4036, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(273, 4096, F2, 13) (dual of [4096, 4023, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(261, 4096, F2, 11) (dual of [4096, 4035, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(213, 14, F2, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,2)), using
- dual of repetition code with length 14 [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 X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(274, 4110, F2, 13) (dual of [4110, 4036, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(275, 4111, F2, 13) (dual of [4111, 4036, 14]-code), using
- net defined by OOA [i] based on linear OOA(275, 685, F2, 13, 13) (dual of [(685, 13), 8830, 14]-NRT-code), using
(65, 78, 1325)-Net over F2 — Digital
Digital (65, 78, 1325)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(278, 1325, F2, 3, 13) (dual of [(1325, 3), 3897, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(278, 1371, F2, 3, 13) (dual of [(1371, 3), 4035, 14]-NRT-code), using
- 21 times duplication [i] based on linear OOA(277, 1371, F2, 3, 13) (dual of [(1371, 3), 4036, 14]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(274, 1370, F2, 3, 13) (dual of [(1370, 3), 4036, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(274, 4110, F2, 13) (dual of [4110, 4036, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(273, 4096, F2, 13) (dual of [4096, 4023, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(261, 4096, F2, 11) (dual of [4096, 4035, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(213, 14, F2, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,2)), using
- dual of repetition code with length 14 [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 X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- OOA 3-folding [i] based on linear OA(274, 4110, F2, 13) (dual of [4110, 4036, 14]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(274, 1370, F2, 3, 13) (dual of [(1370, 3), 4036, 14]-NRT-code), using
- 21 times duplication [i] based on linear OOA(277, 1371, F2, 3, 13) (dual of [(1371, 3), 4036, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(278, 1371, F2, 3, 13) (dual of [(1371, 3), 4035, 14]-NRT-code), using
(65, 78, 21840)-Net in Base 2 — Upper bound on s
There is no (65, 78, 21841)-net in base 2, because
- 1 times m-reduction [i] would yield (65, 77, 21841)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 151118 053348 277892 762040 > 277 [i]