Best Known (195−31, 195, s)-Nets in Base 2
(195−31, 195, 490)-Net over F2 — Constructive and digital
Digital (164, 195, 490)-net over F2, using
- t-expansion [i] based on digital (163, 195, 490)-net over F2, using
- trace code for nets [i] based on digital (7, 39, 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, 39, 98)-net over F32, using
(195−31, 195, 1382)-Net over F2 — Digital
Digital (164, 195, 1382)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2195, 1382, F2, 3, 31) (dual of [(1382, 3), 3951, 32]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2195, 4146, F2, 31) (dual of [4146, 3951, 32]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2193, 4144, F2, 31) (dual of [4144, 3951, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- linear OA(2181, 4096, F2, 31) (dual of [4096, 3915, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2145, 4096, F2, 25) (dual of [4096, 3951, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(30) ⊂ Ce(24) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(2193, 4144, F2, 31) (dual of [4144, 3951, 32]-code), using
- OOA 3-folding [i] based on linear OA(2195, 4146, F2, 31) (dual of [4146, 3951, 32]-code), using
(195−31, 195, 50222)-Net in Base 2 — Upper bound on s
There is no (164, 195, 50223)-net in base 2, because
- 1 times m-reduction [i] would yield (164, 194, 50223)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 25112 652175 392805 449702 308749 928545 884537 068222 716071 081206 > 2194 [i]