Best Known (214, 214+27, s)-Nets in Base 2
(214, 214+27, 20167)-Net over F2 — Constructive and digital
Digital (214, 241, 20167)-net over F2, using
- net defined by OOA [i] based on linear OOA(2241, 20167, F2, 27, 27) (dual of [(20167, 27), 544268, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2241, 262172, F2, 27) (dual of [262172, 261931, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2241, 262176, F2, 27) (dual of [262176, 261935, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- linear OA(2235, 262144, F2, 27) (dual of [262144, 261909, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2199, 262144, F2, 23) (dual of [262144, 261945, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2241, 262176, F2, 27) (dual of [262176, 261935, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2241, 262172, F2, 27) (dual of [262172, 261931, 28]-code), using
(214, 214+27, 37453)-Net over F2 — Digital
Digital (214, 241, 37453)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2241, 37453, F2, 7, 27) (dual of [(37453, 7), 261930, 28]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2241, 262171, F2, 27) (dual of [262171, 261930, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2241, 262176, F2, 27) (dual of [262176, 261935, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- linear OA(2235, 262144, F2, 27) (dual of [262144, 261909, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2199, 262144, F2, 23) (dual of [262144, 261945, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2241, 262176, F2, 27) (dual of [262176, 261935, 28]-code), using
- OOA 7-folding [i] based on linear OA(2241, 262171, F2, 27) (dual of [262171, 261930, 28]-code), using
(214, 214+27, 2045874)-Net in Base 2 — Upper bound on s
There is no (214, 241, 2045875)-net in base 2, because
- 1 times m-reduction [i] would yield (214, 240, 2045875)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1 766852 748147 822965 353228 587766 114885 958155 410405 345511 544316 953959 397376 > 2240 [i]