Best Known (206, 206+39, s)-Nets in Base 2
(206, 206+39, 520)-Net over F2 — Constructive and digital
Digital (206, 245, 520)-net over F2, using
- t-expansion [i] based on digital (205, 245, 520)-net over F2, using
- trace code for nets [i] based on digital (9, 49, 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, 49, 104)-net over F32, using
(206, 206+39, 1485)-Net over F2 — Digital
Digital (206, 245, 1485)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2245, 1485, F2, 2, 39) (dual of [(1485, 2), 2725, 40]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2245, 2074, F2, 2, 39) (dual of [(2074, 2), 3903, 40]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(2241, 2072, F2, 2, 39) (dual of [(2072, 2), 3903, 40]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2241, 4144, F2, 39) (dual of [4144, 3903, 40]-code), using
- construction X applied to Ce(38) ⊂ Ce(32) [i] based on
- linear OA(2229, 4096, F2, 39) (dual of [4096, 3867, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(2193, 4096, F2, 33) (dual of [4096, 3903, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(212, 48, F2, 5) (dual of [48, 36, 6]-code), using
- adding a parity check bit [i] based on linear OA(211, 47, F2, 4) (dual of [47, 36, 5]-code), using
- extracting embedded orthogonal array [i] based on digital (7, 11, 47)-net over F2, using
- adding a parity check bit [i] based on linear OA(211, 47, F2, 4) (dual of [47, 36, 5]-code), using
- construction X applied to Ce(38) ⊂ Ce(32) [i] based on
- OOA 2-folding [i] based on linear OA(2241, 4144, F2, 39) (dual of [4144, 3903, 40]-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(2241, 2072, F2, 2, 39) (dual of [(2072, 2), 3903, 40]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2245, 2074, F2, 2, 39) (dual of [(2074, 2), 3903, 40]-NRT-code), using
(206, 206+39, 58192)-Net in Base 2 — Upper bound on s
There is no (206, 245, 58193)-net in base 2, because
- 1 times m-reduction [i] would yield (206, 244, 58193)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 28 276005 729639 382374 712827 975301 118250 067222 503576 555157 621442 303525 576232 > 2244 [i]