Best Known (172, 172+33, s)-Nets in Base 2
(172, 172+33, 490)-Net over F2 — Constructive and digital
Digital (172, 205, 490)-net over F2, using
- t-expansion [i] based on digital (171, 205, 490)-net over F2, using
- trace code for nets [i] based on digital (7, 41, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 7 and N(F) ≥ 98, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- trace code for nets [i] based on digital (7, 41, 98)-net over F32, using
(172, 172+33, 1381)-Net over F2 — Digital
Digital (172, 205, 1381)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2205, 1381, F2, 3, 33) (dual of [(1381, 3), 3938, 34]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2205, 4143, F2, 33) (dual of [4143, 3938, 34]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(2205, 4144, F2, 33) (dual of [4144, 3939, 34]-code), using
- OOA 3-folding [i] based on linear OA(2205, 4143, F2, 33) (dual of [4143, 3938, 34]-code), using
(172, 172+33, 46822)-Net in Base 2 — Upper bound on s
There is no (172, 205, 46823)-net in base 2, because
- 1 times m-reduction [i] would yield (172, 204, 46823)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 25 716626 839122 206001 367375 826034 618374 663201 201687 273637 175554 > 2204 [i]