Best Known (159, 159+25, s)-Nets in Base 2
(159, 159+25, 2732)-Net over F2 — Constructive and digital
Digital (159, 184, 2732)-net over F2, using
- 21 times duplication [i] based on digital (158, 183, 2732)-net over F2, using
- net defined by OOA [i] based on linear OOA(2183, 2732, F2, 25, 25) (dual of [(2732, 25), 68117, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2183, 32785, F2, 25) (dual of [32785, 32602, 26]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2182, 32784, F2, 25) (dual of [32784, 32602, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(2181, 32768, F2, 25) (dual of [32768, 32587, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2166, 32768, F2, 23) (dual of [32768, 32602, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(24) ⊂ Ce(22) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2182, 32784, F2, 25) (dual of [32784, 32602, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2183, 32785, F2, 25) (dual of [32785, 32602, 26]-code), using
- net defined by OOA [i] based on linear OOA(2183, 2732, F2, 25, 25) (dual of [(2732, 25), 68117, 26]-NRT-code), using
(159, 159+25, 5464)-Net over F2 — Digital
Digital (159, 184, 5464)-net over F2, using
- 22 times duplication [i] based on digital (157, 182, 5464)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2182, 5464, F2, 6, 25) (dual of [(5464, 6), 32602, 26]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2182, 32784, F2, 25) (dual of [32784, 32602, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(2181, 32768, F2, 25) (dual of [32768, 32587, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2166, 32768, F2, 23) (dual of [32768, 32602, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(24) ⊂ Ce(22) [i] based on
- OOA 6-folding [i] based on linear OA(2182, 32784, F2, 25) (dual of [32784, 32602, 26]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2182, 5464, F2, 6, 25) (dual of [(5464, 6), 32602, 26]-NRT-code), using
(159, 159+25, 206078)-Net in Base 2 — Upper bound on s
There is no (159, 184, 206079)-net in base 2, because
- 1 times m-reduction [i] would yield (159, 183, 206079)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 12 260563 107619 176323 447191 517858 008282 366615 503270 415701 > 2183 [i]