Best Known (210, 210+38, s)-Nets in Base 2
(210, 210+38, 624)-Net over F2 — Constructive and digital
Digital (210, 248, 624)-net over F2, using
- 22 times duplication [i] based on digital (208, 246, 624)-net over F2, using
- trace code for nets [i] based on digital (3, 41, 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, 41, 104)-net over F64, using
(210, 210+38, 2051)-Net over F2 — Digital
Digital (210, 248, 2051)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2248, 2051, F2, 4, 38) (dual of [(2051, 4), 7956, 39]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2248, 8204, F2, 38) (dual of [8204, 7956, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(2248, 8205, F2, 38) (dual of [8205, 7957, 39]-code), using
- 1 times truncation [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
- 1 times truncation [i] based on linear OA(2249, 8206, F2, 39) (dual of [8206, 7957, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(2248, 8205, F2, 38) (dual of [8205, 7957, 39]-code), using
- OOA 4-folding [i] based on linear OA(2248, 8204, F2, 38) (dual of [8204, 7956, 39]-code), using
(210, 210+38, 67339)-Net in Base 2 — Upper bound on s
There is no (210, 248, 67340)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 452 401631 479939 355462 220577 481372 699852 845525 630549 765981 468638 734032 153488 > 2248 [i]