Best Known (227−38, 227, s)-Nets in Base 2
(227−38, 227, 490)-Net over F2 — Constructive and digital
Digital (189, 227, 490)-net over F2, using
- 22 times duplication [i] based on digital (187, 225, 490)-net over F2, using
- trace code for nets [i] based on digital (7, 45, 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, 45, 98)-net over F32, using
(227−38, 227, 1060)-Net over F2 — Digital
Digital (189, 227, 1060)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2227, 1060, F2, 38) (dual of [1060, 833, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(2227, 2098, F2, 38) (dual of [2098, 1871, 39]-code), using
- 6 times code embedding in larger space [i] based on linear OA(2221, 2092, F2, 38) (dual of [2092, 1871, 39]-code), using
- 1 times truncation [i] based on linear OA(2222, 2093, F2, 39) (dual of [2093, 1871, 40]-code), using
- construction X applied to Ce(38) ⊂ Ce(32) [i] based on
- linear OA(2210, 2048, F2, 39) (dual of [2048, 1838, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(2177, 2048, F2, 33) (dual of [2048, 1871, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [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(38) ⊂ Ce(32) [i] based on
- 1 times truncation [i] based on linear OA(2222, 2093, F2, 39) (dual of [2093, 1871, 40]-code), using
- 6 times code embedding in larger space [i] based on linear OA(2221, 2092, F2, 38) (dual of [2092, 1871, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(2227, 2098, F2, 38) (dual of [2098, 1871, 39]-code), using
(227−38, 227, 31285)-Net in Base 2 — Upper bound on s
There is no (189, 227, 31286)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 215 742036 372055 574937 225051 970365 651896 095460 031665 041947 745512 237918 > 2227 [i]