Best Known (237−40, 237, s)-Nets in Base 2
(237−40, 237, 490)-Net over F2 — Constructive and digital
Digital (197, 237, 490)-net over F2, using
- 22 times duplication [i] based on digital (195, 235, 490)-net over F2, using
- trace code for nets [i] based on digital (7, 47, 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, 47, 98)-net over F32, using
(237−40, 237, 1062)-Net over F2 — Digital
Digital (197, 237, 1062)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2237, 1062, F2, 40) (dual of [1062, 825, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(2237, 2097, F2, 40) (dual of [2097, 1860, 41]-code), using
- 5 times code embedding in larger space [i] based on linear OA(2232, 2092, F2, 40) (dual of [2092, 1860, 41]-code), using
- 1 times truncation [i] based on linear OA(2233, 2093, F2, 41) (dual of [2093, 1860, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(34) [i] based on
- linear OA(2221, 2048, F2, 41) (dual of [2048, 1827, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(2188, 2048, F2, 35) (dual of [2048, 1860, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(212, 45, F2, 5) (dual of [45, 33, 6]-code), using
- discarding factors / shortening the dual code based on 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
- discarding factors / shortening the dual code based on linear OA(212, 48, F2, 5) (dual of [48, 36, 6]-code), using
- construction X applied to Ce(40) ⊂ Ce(34) [i] based on
- 1 times truncation [i] based on linear OA(2233, 2093, F2, 41) (dual of [2093, 1860, 42]-code), using
- 5 times code embedding in larger space [i] based on linear OA(2232, 2092, F2, 40) (dual of [2092, 1860, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(2237, 2097, F2, 40) (dual of [2097, 1860, 41]-code), using
(237−40, 237, 30626)-Net in Base 2 — Upper bound on s
There is no (197, 237, 30627)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 220964 819293 276843 555068 176311 348977 386994 453140 511587 148159 205280 215336 > 2237 [i]