Best Known (54, 54+9, s)-Nets in Base 2
(54, 54+9, 8196)-Net over F2 — Constructive and digital
Digital (54, 63, 8196)-net over F2, using
- 21 times duplication [i] based on digital (53, 62, 8196)-net over F2, using
- net defined by OOA [i] based on linear OOA(262, 8196, F2, 9, 9) (dual of [(8196, 9), 73702, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(262, 8196, F2, 8, 9) (dual of [(8196, 8), 65506, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(262, 32785, F2, 9) (dual of [32785, 32723, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(261, 32768, F2, 9) (dual of [32768, 32707, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(246, 32768, F2, 7) (dual of [32768, 32722, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(216, 17, F2, 15) (dual of [17, 1, 16]-code), using
- strength reduction [i] based on linear OA(216, 17, F2, 16) (dual of [17, 1, 17]-code or 17-arc in PG(15,2)), using
- dual of repetition code with length 17 [i]
- strength reduction [i] based on linear OA(216, 17, F2, 16) (dual of [17, 1, 17]-code or 17-arc in PG(15,2)), using
- linear OA(21, 17, F2, 1) (dual of [17, 16, 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(8) ⊂ Ce(6) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(262, 32785, F2, 9) (dual of [32785, 32723, 10]-code), using
- appending kth column [i] based on linear OOA(262, 8196, F2, 8, 9) (dual of [(8196, 8), 65506, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(262, 8196, F2, 9, 9) (dual of [(8196, 9), 73702, 10]-NRT-code), using
(54, 54+9, 10664)-Net over F2 — Digital
Digital (54, 63, 10664)-net over F2, using
- net defined by OOA [i] based on linear OOA(263, 10664, F2, 9, 9) (dual of [(10664, 9), 95913, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(263, 10664, F2, 8, 9) (dual of [(10664, 8), 85249, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(263, 10664, F2, 3, 9) (dual of [(10664, 3), 31929, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(263, 10928, F2, 3, 9) (dual of [(10928, 3), 32721, 10]-NRT-code), using
- 21 times duplication [i] based on linear OOA(262, 10928, F2, 3, 9) (dual of [(10928, 3), 32722, 10]-NRT-code), using
- OOA 3-folding [i] based on linear OA(262, 32784, F2, 9) (dual of [32784, 32722, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(261, 32768, F2, 9) (dual of [32768, 32707, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(246, 32768, F2, 7) (dual of [32768, 32722, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(21, 16, F2, 1) (dual of [16, 15, 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(8) ⊂ Ce(6) [i] based on
- OOA 3-folding [i] based on linear OA(262, 32784, F2, 9) (dual of [32784, 32722, 10]-code), using
- 21 times duplication [i] based on linear OOA(262, 10928, F2, 3, 9) (dual of [(10928, 3), 32722, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(263, 10928, F2, 3, 9) (dual of [(10928, 3), 32721, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(263, 10664, F2, 3, 9) (dual of [(10664, 3), 31929, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(263, 10664, F2, 8, 9) (dual of [(10664, 8), 85249, 10]-NRT-code), using
(54, 54+9, 102563)-Net in Base 2 — Upper bound on s
There is no (54, 63, 102564)-net in base 2, because
- 1 times m-reduction [i] would yield (54, 62, 102564)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 4 611707 031230 006632 > 262 [i]