Best Known (224, 224+35, s)-Nets in Base 2
(224, 224+35, 1928)-Net over F2 — Constructive and digital
Digital (224, 259, 1928)-net over F2, using
- 22 times duplication [i] based on digital (222, 257, 1928)-net over F2, using
- net defined by OOA [i] based on linear OOA(2257, 1928, F2, 35, 35) (dual of [(1928, 35), 67223, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(2257, 32777, F2, 35) (dual of [32777, 32520, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(2257, 32784, F2, 35) (dual of [32784, 32527, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(32) [i] based on
- linear OA(2256, 32768, F2, 35) (dual of [32768, 32512, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2241, 32768, F2, 33) (dual of [32768, 32527, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [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(34) ⊂ Ce(32) [i] based on
- discarding factors / shortening the dual code based on linear OA(2257, 32784, F2, 35) (dual of [32784, 32527, 36]-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(2257, 32777, F2, 35) (dual of [32777, 32520, 36]-code), using
- net defined by OOA [i] based on linear OOA(2257, 1928, F2, 35, 35) (dual of [(1928, 35), 67223, 36]-NRT-code), using
(224, 224+35, 5464)-Net over F2 — Digital
Digital (224, 259, 5464)-net over F2, using
- 22 times duplication [i] based on digital (222, 257, 5464)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2257, 5464, F2, 6, 35) (dual of [(5464, 6), 32527, 36]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2257, 32784, F2, 35) (dual of [32784, 32527, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(32) [i] based on
- linear OA(2256, 32768, F2, 35) (dual of [32768, 32512, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2241, 32768, F2, 33) (dual of [32768, 32527, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [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(34) ⊂ Ce(32) [i] based on
- OOA 6-folding [i] based on linear OA(2257, 32784, F2, 35) (dual of [32784, 32527, 36]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2257, 5464, F2, 6, 35) (dual of [(5464, 6), 32527, 36]-NRT-code), using
(224, 224+35, 265752)-Net in Base 2 — Upper bound on s
There is no (224, 259, 265753)-net in base 2, because
- 1 times m-reduction [i] would yield (224, 258, 265753)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 463185 492305 917186 363041 408753 929546 850213 501630 102841 261240 100370 858157 656694 > 2258 [i]