Best Known (231−31, 231, s)-Nets in Base 2
(231−31, 231, 2185)-Net over F2 — Constructive and digital
Digital (200, 231, 2185)-net over F2, using
- 24 times duplication [i] based on digital (196, 227, 2185)-net over F2, using
- net defined by OOA [i] based on linear OOA(2227, 2185, F2, 31, 31) (dual of [(2185, 31), 67508, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2227, 32776, F2, 31) (dual of [32776, 32549, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2227, 32784, F2, 31) (dual of [32784, 32557, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(2226, 32768, F2, 31) (dual of [32768, 32542, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2211, 32768, F2, 29) (dual of [32768, 32557, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(2227, 32784, F2, 31) (dual of [32784, 32557, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2227, 32776, F2, 31) (dual of [32776, 32549, 32]-code), using
- net defined by OOA [i] based on linear OOA(2227, 2185, F2, 31, 31) (dual of [(2185, 31), 67508, 32]-NRT-code), using
(231−31, 231, 5464)-Net over F2 — Digital
Digital (200, 231, 5464)-net over F2, using
- 24 times duplication [i] based on digital (196, 227, 5464)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2227, 5464, F2, 6, 31) (dual of [(5464, 6), 32557, 32]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2227, 32784, F2, 31) (dual of [32784, 32557, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(2226, 32768, F2, 31) (dual of [32768, 32542, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2211, 32768, F2, 29) (dual of [32768, 32557, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(30) ⊂ Ce(28) [i] based on
- OOA 6-folding [i] based on linear OA(2227, 32784, F2, 31) (dual of [32784, 32557, 32]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2227, 5464, F2, 6, 31) (dual of [(5464, 6), 32557, 32]-NRT-code), using
(231−31, 231, 265169)-Net in Base 2 — Upper bound on s
There is no (200, 231, 265170)-net in base 2, because
- 1 times m-reduction [i] would yield (200, 230, 265170)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1725 466962 391257 212658 369174 941671 519117 168897 011549 882178 478809 919528 > 2230 [i]