Best Known (51, 68, s)-Nets in Base 2
(51, 68, 87)-Net over F2 — Constructive and digital
Digital (51, 68, 87)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (13, 21, 39)-net over F2, using
- digital (30, 47, 48)-net over F2, using
- 1 times m-reduction [i] based on digital (30, 48, 48)-net over F2, using
(51, 68, 137)-Net over F2 — Digital
Digital (51, 68, 137)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(268, 137, F2, 2, 17) (dual of [(137, 2), 206, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(268, 274, F2, 17) (dual of [274, 206, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(268, 275, F2, 17) (dual of [275, 207, 18]-code), using
- adding a parity check bit [i] based on linear OA(267, 274, F2, 16) (dual of [274, 207, 17]-code), using
- construction XX applied to C1 = C([253,12]), C2 = C([1,14]), C3 = C1 + C2 = C([1,12]), 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(256, 255, F2, 14) (dual of [255, 199, 15]-code), using the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(248, 255, F2, 12) (dual of [255, 207, 13]-code), using the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(21, 10, F2, 1) (dual of [10, 9, 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), 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 (see above)
- construction XX applied to C1 = C([253,12]), C2 = C([1,14]), C3 = C1 + C2 = C([1,12]), and C∩ = C1 ∩ C2 = C([253,14]) [i] based on
- adding a parity check bit [i] based on linear OA(267, 274, F2, 16) (dual of [274, 207, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(268, 275, F2, 17) (dual of [275, 207, 18]-code), using
- OOA 2-folding [i] based on linear OA(268, 274, F2, 17) (dual of [274, 206, 18]-code), using
(51, 68, 1238)-Net in Base 2 — Upper bound on s
There is no (51, 68, 1239)-net in base 2, because
- 1 times m-reduction [i] would yield (51, 67, 1239)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 148 258423 220776 694402 > 267 [i]