Best Known (163, 163+30, s)-Nets in Base 2
(163, 163+30, 490)-Net over F2 — Constructive and digital
Digital (163, 193, 490)-net over F2, using
- 2 times m-reduction [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
(163, 163+30, 1433)-Net over F2 — Digital
Digital (163, 193, 1433)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2193, 1433, F2, 2, 30) (dual of [(1433, 2), 2673, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2193, 2072, F2, 2, 30) (dual of [(2072, 2), 3951, 31]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2193, 4144, F2, 30) (dual of [4144, 3951, 31]-code), using
- strength reduction [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
- strength reduction [i] based on linear OA(2193, 4144, F2, 31) (dual of [4144, 3951, 32]-code), using
- OOA 2-folding [i] based on linear OA(2193, 4144, F2, 30) (dual of [4144, 3951, 31]-code), using
- discarding factors / shortening the dual code based on linear OOA(2193, 2072, F2, 2, 30) (dual of [(2072, 2), 3951, 31]-NRT-code), using
(163, 163+30, 47953)-Net in Base 2 — Upper bound on s
There is no (163, 193, 47954)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 12556 260165 953992 191339 847554 878766 351729 855309 568300 057448 > 2193 [i]