Best Known (222, 222+38, s)-Nets in Base 2
(222, 222+38, 768)-Net over F2 — Constructive and digital
Digital (222, 260, 768)-net over F2, using
- 22 times duplication [i] based on digital (220, 258, 768)-net over F2, using
- trace code for nets [i] based on digital (5, 43, 128)-net over F64, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 5 and N(F) ≥ 128, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- trace code for nets [i] based on digital (5, 43, 128)-net over F64, using
(222, 222+38, 2503)-Net over F2 — Digital
Digital (222, 260, 2503)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2260, 2503, F2, 3, 38) (dual of [(2503, 3), 7249, 39]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2260, 2747, F2, 3, 38) (dual of [(2747, 3), 7981, 39]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2260, 8241, F2, 38) (dual of [8241, 7981, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(2260, 8243, F2, 38) (dual of [8243, 7983, 39]-code), using
- construction X applied to Ce(38) ⊂ Ce(32) [i] based on
- linear OA(2248, 8192, F2, 39) (dual of [8192, 7944, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [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(212, 51, F2, 4) (dual of [51, 39, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(212, 63, F2, 4) (dual of [63, 51, 5]-code), using
- the primitive narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(212, 63, F2, 4) (dual of [63, 51, 5]-code), using
- construction X applied to Ce(38) ⊂ Ce(32) [i] based on
- discarding factors / shortening the dual code based on linear OA(2260, 8243, F2, 38) (dual of [8243, 7983, 39]-code), using
- OOA 3-folding [i] based on linear OA(2260, 8241, F2, 38) (dual of [8241, 7981, 39]-code), using
- discarding factors / shortening the dual code based on linear OOA(2260, 2747, F2, 3, 38) (dual of [(2747, 3), 7981, 39]-NRT-code), using
(222, 222+38, 104341)-Net in Base 2 — Upper bound on s
There is no (222, 260, 104342)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 1 852862 084190 873762 632059 062582 478853 627839 912555 312597 285193 240816 963922 149154 > 2260 [i]