Best Known (215, 215+31, s)-Nets in Base 2
(215, 215+31, 4370)-Net over F2 — Constructive and digital
Digital (215, 246, 4370)-net over F2, using
- 24 times duplication [i] based on digital (211, 242, 4370)-net over F2, using
- net defined by OOA [i] based on linear OOA(2242, 4370, F2, 31, 31) (dual of [(4370, 31), 135228, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2242, 65551, F2, 31) (dual of [65551, 65309, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2242, 65553, F2, 31) (dual of [65553, 65311, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(2241, 65536, F2, 31) (dual of [65536, 65295, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2225, 65536, F2, 29) (dual of [65536, 65311, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- 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 X applied to Ce(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(2242, 65553, F2, 31) (dual of [65553, 65311, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2242, 65551, F2, 31) (dual of [65551, 65309, 32]-code), using
- net defined by OOA [i] based on linear OOA(2242, 4370, F2, 31, 31) (dual of [(4370, 31), 135228, 32]-NRT-code), using
(215, 215+31, 10003)-Net over F2 — Digital
Digital (215, 246, 10003)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2246, 10003, F2, 6, 31) (dual of [(10003, 6), 59772, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2246, 10926, F2, 6, 31) (dual of [(10926, 6), 65310, 32]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2245, 10926, F2, 6, 31) (dual of [(10926, 6), 65311, 32]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2245, 65556, F2, 31) (dual of [65556, 65311, 32]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2242, 65553, F2, 31) (dual of [65553, 65311, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(2241, 65536, F2, 31) (dual of [65536, 65295, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2225, 65536, F2, 29) (dual of [65536, 65311, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- 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 X applied to Ce(30) ⊂ Ce(28) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(2242, 65553, F2, 31) (dual of [65553, 65311, 32]-code), using
- OOA 6-folding [i] based on linear OA(2245, 65556, F2, 31) (dual of [65556, 65311, 32]-code), using
- 21 times duplication [i] based on linear OOA(2245, 10926, F2, 6, 31) (dual of [(10926, 6), 65311, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2246, 10926, F2, 6, 31) (dual of [(10926, 6), 65310, 32]-NRT-code), using
(215, 215+31, 530361)-Net in Base 2 — Upper bound on s
There is no (215, 246, 530362)-net in base 2, because
- 1 times m-reduction [i] would yield (215, 245, 530362)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 56 540103 407305 403111 063022 087712 545662 634420 527719 194166 717415 077188 548168 > 2245 [i]