Best Known (199, 224, s)-Nets in Base 2
(199, 224, 21848)-Net over F2 — Constructive and digital
Digital (199, 224, 21848)-net over F2, using
- 21 times duplication [i] based on digital (198, 223, 21848)-net over F2, using
- net defined by OOA [i] based on linear OOA(2223, 21848, F2, 25, 25) (dual of [(21848, 25), 545977, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2223, 262177, F2, 25) (dual of [262177, 261954, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(2217, 262145, F2, 25) (dual of [262145, 261928, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 236−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2181, 262145, F2, 21) (dual of [262145, 261964, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 236−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [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 C([0,12]) ⊂ C([0,10]) [i] based on
- OOA 12-folding and stacking with additional row [i] based on linear OA(2223, 262177, F2, 25) (dual of [262177, 261954, 26]-code), using
- net defined by OOA [i] based on linear OOA(2223, 21848, F2, 25, 25) (dual of [(21848, 25), 545977, 26]-NRT-code), using
(199, 224, 37455)-Net over F2 — Digital
Digital (199, 224, 37455)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2224, 37455, F2, 7, 25) (dual of [(37455, 7), 261961, 26]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2224, 262185, F2, 25) (dual of [262185, 261961, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2224, 262188, F2, 25) (dual of [262188, 261964, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(2217, 262145, F2, 25) (dual of [262145, 261928, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 236−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2181, 262145, F2, 21) (dual of [262145, 261964, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 236−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(27, 43, F2, 3) (dual of [43, 36, 4]-code or 43-cap in PG(6,2)), using
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 4 [i]
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2224, 262188, F2, 25) (dual of [262188, 261964, 26]-code), using
- OOA 7-folding [i] based on linear OA(2224, 262185, F2, 25) (dual of [262185, 261961, 26]-code), using
(199, 224, 2077296)-Net in Base 2 — Upper bound on s
There is no (199, 224, 2077297)-net in base 2, because
- 1 times m-reduction [i] would yield (199, 223, 2077297)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 13 480005 411152 026980 568050 571087 077903 067715 660617 885844 800692 148468 > 2223 [i]