Best Known (223−34, 223, s)-Nets in Base 2
(223−34, 223, 624)-Net over F2 — Constructive and digital
Digital (189, 223, 624)-net over F2, using
- 21 times duplication [i] based on digital (188, 222, 624)-net over F2, using
- trace code for nets [i] based on digital (3, 37, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- trace code for nets [i] based on digital (3, 37, 104)-net over F64, using
(223−34, 223, 2051)-Net over F2 — Digital
Digital (189, 223, 2051)-net over F2, using
- 21 times duplication [i] based on digital (188, 222, 2051)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2222, 2051, F2, 4, 34) (dual of [(2051, 4), 7982, 35]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2222, 8204, F2, 34) (dual of [8204, 7982, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(2222, 8205, F2, 34) (dual of [8205, 7983, 35]-code), using
- 1 times truncation [i] based on linear OA(2223, 8206, F2, 35) (dual of [8206, 7983, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(32) [i] based on
- linear OA(2222, 8192, F2, 35) (dual of [8192, 7970, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2209, 8192, F2, 33) (dual of [8192, 7983, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(34) ⊂ Ce(32) [i] based on
- 1 times truncation [i] based on linear OA(2223, 8206, F2, 35) (dual of [8206, 7983, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(2222, 8205, F2, 34) (dual of [8205, 7983, 35]-code), using
- OOA 4-folding [i] based on linear OA(2222, 8204, F2, 34) (dual of [8204, 7982, 35]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2222, 2051, F2, 4, 34) (dual of [(2051, 4), 7982, 35]-NRT-code), using
(223−34, 223, 63764)-Net in Base 2 — Upper bound on s
There is no (189, 223, 63765)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 13 481096 151441 399480 688237 266638 574608 918243 210385 625610 491797 399438 > 2223 [i]