Best Known (214−33, 214, s)-Nets in Base 2
(214−33, 214, 520)-Net over F2 — Constructive and digital
Digital (181, 214, 520)-net over F2, using
- 1 times m-reduction [i] based on digital (181, 215, 520)-net over F2, using
- trace code for nets [i] based on digital (9, 43, 104)-net over F32, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- trace code for nets [i] based on digital (9, 43, 104)-net over F32, using
(214−33, 214, 2005)-Net over F2 — Digital
Digital (181, 214, 2005)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2214, 2005, F2, 4, 33) (dual of [(2005, 4), 7806, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2214, 2052, F2, 4, 33) (dual of [(2052, 4), 7994, 34]-NRT-code), using
- 22 times duplication [i] based on linear OOA(2212, 2052, F2, 4, 33) (dual of [(2052, 4), 7996, 34]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2212, 8208, F2, 33) (dual of [8208, 7996, 34]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2210, 8206, F2, 33) (dual of [8206, 7996, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(30) [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]
- linear OA(2196, 8192, F2, 31) (dual of [8192, 7996, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 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
- 2 times code embedding in larger space [i] based on linear OA(2210, 8206, F2, 33) (dual of [8206, 7996, 34]-code), using
- OOA 4-folding [i] based on linear OA(2212, 8208, F2, 33) (dual of [8208, 7996, 34]-code), using
- 22 times duplication [i] based on linear OOA(2212, 2052, F2, 4, 33) (dual of [(2052, 4), 7996, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2214, 2052, F2, 4, 33) (dual of [(2052, 4), 7994, 34]-NRT-code), using
(214−33, 214, 69159)-Net in Base 2 — Upper bound on s
There is no (181, 214, 69160)-net in base 2, because
- 1 times m-reduction [i] would yield (181, 213, 69160)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 13164 989291 621325 515669 294124 252962 937177 694066 994475 261219 203550 > 2213 [i]