Best Known (227−33, 227, s)-Nets in Base 2
(227−33, 227, 1024)-Net over F2 — Constructive and digital
Digital (194, 227, 1024)-net over F2, using
- 22 times duplication [i] based on digital (192, 225, 1024)-net over F2, using
- net defined by OOA [i] based on linear OOA(2225, 1024, F2, 33, 33) (dual of [(1024, 33), 33567, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(2225, 16385, F2, 33) (dual of [16385, 16160, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16385 | 228−1, 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(2225, 16385, F2, 33) (dual of [16385, 16160, 34]-code), using
- net defined by OOA [i] based on linear OOA(2225, 1024, F2, 33, 33) (dual of [(1024, 33), 33567, 34]-NRT-code), using
(227−33, 227, 3223)-Net over F2 — Digital
Digital (194, 227, 3223)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2227, 3223, F2, 5, 33) (dual of [(3223, 5), 15888, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2227, 3280, F2, 5, 33) (dual of [(3280, 5), 16173, 34]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2227, 16400, F2, 33) (dual of [16400, 16173, 34]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2226, 16399, F2, 33) (dual of [16399, 16173, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- linear OA(2225, 16384, F2, 33) (dual of [16384, 16159, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(2211, 16384, F2, 31) (dual of [16384, 16173, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(21, 15, F2, 1) (dual of [15, 14, 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(32) ⊂ Ce(30) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2226, 16399, F2, 33) (dual of [16399, 16173, 34]-code), using
- OOA 5-folding [i] based on linear OA(2227, 16400, F2, 33) (dual of [16400, 16173, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(2227, 3280, F2, 5, 33) (dual of [(3280, 5), 16173, 34]-NRT-code), using
(227−33, 227, 121479)-Net in Base 2 — Upper bound on s
There is no (194, 227, 121480)-net in base 2, because
- 1 times m-reduction [i] would yield (194, 226, 121480)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 107 845220 064416 005193 732310 254948 130481 836621 095332 866773 908715 311288 > 2226 [i]