Best Known (85−13, 85, s)-Nets in Base 2
(85−13, 85, 2730)-Net over F2 — Constructive and digital
Digital (72, 85, 2730)-net over F2, using
- net defined by OOA [i] based on linear OOA(285, 2730, F2, 13, 13) (dual of [(2730, 13), 35405, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(285, 16381, F2, 13) (dual of [16381, 16296, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(285, 16384, F2, 13) (dual of [16384, 16299, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(285, 16384, F2, 13) (dual of [16384, 16299, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(285, 16381, F2, 13) (dual of [16381, 16296, 14]-code), using
(85−13, 85, 4096)-Net over F2 — Digital
Digital (72, 85, 4096)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(285, 4096, F2, 4, 13) (dual of [(4096, 4), 16299, 14]-NRT-code), using
- OOA 4-folding [i] based on linear OA(285, 16384, F2, 13) (dual of [16384, 16299, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- OOA 4-folding [i] based on linear OA(285, 16384, F2, 13) (dual of [16384, 16299, 14]-code), using
(85−13, 85, 49041)-Net in Base 2 — Upper bound on s
There is no (72, 85, 49042)-net in base 2, because
- 1 times m-reduction [i] would yield (72, 84, 49042)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 19 343190 190914 556102 756600 > 284 [i]