Best Known (176, 209, s)-Nets in Base 2
(176, 209, 512)-Net over F2 — Constructive and digital
Digital (176, 209, 512)-net over F2, using
- net defined by OOA [i] based on linear OOA(2209, 512, F2, 33, 33) (dual of [(512, 33), 16687, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(2209, 8193, F2, 33) (dual of [8193, 7984, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 8193 | 226−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- OOA 16-folding and stacking with additional row [i] based on linear OA(2209, 8193, F2, 33) (dual of [8193, 7984, 34]-code), using
(176, 209, 1766)-Net over F2 — Digital
Digital (176, 209, 1766)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2209, 1766, F2, 4, 33) (dual of [(1766, 4), 6855, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2209, 2048, F2, 4, 33) (dual of [(2048, 4), 7983, 34]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2209, 8192, F2, 33) (dual of [8192, 7983, 34]-code), using
- an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- OOA 4-folding [i] based on linear OA(2209, 8192, F2, 33) (dual of [8192, 7983, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(2209, 2048, F2, 4, 33) (dual of [(2048, 4), 7983, 34]-NRT-code), using
(176, 209, 55685)-Net in Base 2 — Upper bound on s
There is no (176, 209, 55686)-net in base 2, because
- 1 times m-reduction [i] would yield (176, 208, 55686)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 411 384585 995741 160010 187586 176554 899070 957747 511856 352184 187336 > 2208 [i]