Best Known (206−32, 206, s)-Nets in Base 2
(206−32, 206, 520)-Net over F2 — Constructive and digital
Digital (174, 206, 520)-net over F2, using
- 21 times duplication [i] based on digital (173, 205, 520)-net over F2, using
- trace code for nets [i] based on digital (9, 41, 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, 41, 104)-net over F32, using
(206−32, 206, 1488)-Net over F2 — Digital
Digital (174, 206, 1488)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2206, 1488, F2, 2, 32) (dual of [(1488, 2), 2770, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2206, 2072, F2, 2, 32) (dual of [(2072, 2), 3938, 33]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2205, 2072, F2, 2, 32) (dual of [(2072, 2), 3939, 33]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2205, 4144, F2, 32) (dual of [4144, 3939, 33]-code), using
- strength reduction [i] based on linear OA(2205, 4144, F2, 33) (dual of [4144, 3939, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(26) [i] based on
- 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(2157, 4096, F2, 27) (dual of [4096, 3939, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(32) ⊂ Ce(26) [i] based on
- strength reduction [i] based on linear OA(2205, 4144, F2, 33) (dual of [4144, 3939, 34]-code), using
- OOA 2-folding [i] based on linear OA(2205, 4144, F2, 32) (dual of [4144, 3939, 33]-code), using
- 21 times duplication [i] based on linear OOA(2205, 2072, F2, 2, 32) (dual of [(2072, 2), 3939, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2206, 2072, F2, 2, 32) (dual of [(2072, 2), 3938, 33]-NRT-code), using
(206−32, 206, 51062)-Net in Base 2 — Upper bound on s
There is no (174, 206, 51063)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 102 867487 716485 371772 823140 576899 012486 161550 500478 850256 006247 > 2206 [i]