Best Known (214, 214+38, s)-Nets in Base 2
(214, 214+38, 624)-Net over F2 — Constructive and digital
Digital (214, 252, 624)-net over F2, using
- t-expansion [i] based on digital (213, 252, 624)-net over F2, using
- trace code for nets [i] based on digital (3, 42, 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, 42, 104)-net over F64, using
(214, 214+38, 2119)-Net over F2 — Digital
Digital (214, 252, 2119)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2252, 2119, F2, 3, 38) (dual of [(2119, 3), 6105, 39]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2252, 2736, F2, 3, 38) (dual of [(2736, 3), 7956, 39]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2251, 2736, F2, 3, 38) (dual of [(2736, 3), 7957, 39]-NRT-code), using
- strength reduction [i] based on linear OOA(2251, 2736, F2, 3, 39) (dual of [(2736, 3), 7957, 40]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2251, 8208, F2, 39) (dual of [8208, 7957, 40]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2249, 8206, F2, 39) (dual of [8206, 7957, 40]-code), using
- construction X applied to Ce(38) ⊂ Ce(36) [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(2235, 8192, F2, 37) (dual of [8192, 7957, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [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(38) ⊂ Ce(36) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(2249, 8206, F2, 39) (dual of [8206, 7957, 40]-code), using
- OOA 3-folding [i] based on linear OA(2251, 8208, F2, 39) (dual of [8208, 7957, 40]-code), using
- strength reduction [i] based on linear OOA(2251, 2736, F2, 3, 39) (dual of [(2736, 3), 7957, 40]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2251, 2736, F2, 3, 38) (dual of [(2736, 3), 7957, 39]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2252, 2736, F2, 3, 38) (dual of [(2736, 3), 7956, 39]-NRT-code), using
(214, 214+38, 77923)-Net in Base 2 — Upper bound on s
There is no (214, 252, 77924)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 7238 071642 327318 085870 352447 827718 563846 602333 992186 410354 289271 422477 814062 > 2252 [i]