Best Known (50, 67, s)-Nets in Base 2
(50, 67, 85)-Net over F2 — Constructive and digital
Digital (50, 67, 85)-net over F2, using
- (u, u+v)-construction [i] based on
(50, 67, 133)-Net over F2 — Digital
Digital (50, 67, 133)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(267, 133, F2, 2, 17) (dual of [(133, 2), 199, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(267, 136, F2, 2, 17) (dual of [(136, 2), 205, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(267, 272, F2, 17) (dual of [272, 205, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(267, 273, F2, 17) (dual of [273, 206, 18]-code), using
- construction XX applied to C1 = C([253,12]), C2 = C([0,14]), C3 = C1 + C2 = C([0,12]), and C∩ = C1 ∩ C2 = C([253,14]) [i] based on
- linear OA(257, 255, F2, 15) (dual of [255, 198, 16]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,12}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(257, 255, F2, 15) (dual of [255, 198, 16]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(265, 255, F2, 17) (dual of [255, 190, 18]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,14}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(249, 255, F2, 13) (dual of [255, 206, 14]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(21, 9, F2, 1) (dual of [9, 8, 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
- linear OA(21, 9, F2, 1) (dual of [9, 8, 2]-code) (see above)
- construction XX applied to C1 = C([253,12]), C2 = C([0,14]), C3 = C1 + C2 = C([0,12]), and C∩ = C1 ∩ C2 = C([253,14]) [i] based on
- discarding factors / shortening the dual code based on linear OA(267, 273, F2, 17) (dual of [273, 206, 18]-code), using
- OOA 2-folding [i] based on linear OA(267, 272, F2, 17) (dual of [272, 205, 18]-code), using
- discarding factors / shortening the dual code based on linear OOA(267, 136, F2, 2, 17) (dual of [(136, 2), 205, 18]-NRT-code), using
(50, 67, 1134)-Net in Base 2 — Upper bound on s
There is no (50, 67, 1135)-net in base 2, because
- 1 times m-reduction [i] would yield (50, 66, 1135)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 74 014660 462716 040423 > 266 [i]