Best Known (174, 199, s)-Nets in Base 2
(174, 199, 5464)-Net over F2 — Constructive and digital
Digital (174, 199, 5464)-net over F2, using
- net defined by OOA [i] based on linear OOA(2199, 5464, F2, 25, 25) (dual of [(5464, 25), 136401, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2199, 65569, F2, 25) (dual of [65569, 65370, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(2193, 65537, F2, 25) (dual of [65537, 65344, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2161, 65537, F2, 21) (dual of [65537, 65376, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, 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(2199, 65569, F2, 25) (dual of [65569, 65370, 26]-code), using
(174, 199, 10928)-Net over F2 — Digital
Digital (174, 199, 10928)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2199, 10928, F2, 6, 25) (dual of [(10928, 6), 65369, 26]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2199, 65568, F2, 25) (dual of [65568, 65369, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2199, 65569, F2, 25) (dual of [65569, 65370, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(2193, 65537, F2, 25) (dual of [65537, 65344, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2161, 65537, F2, 21) (dual of [65537, 65376, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, 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
- discarding factors / shortening the dual code based on linear OA(2199, 65569, F2, 25) (dual of [65569, 65370, 26]-code), using
- OOA 6-folding [i] based on linear OA(2199, 65568, F2, 25) (dual of [65568, 65369, 26]-code), using
(174, 199, 490163)-Net in Base 2 — Upper bound on s
There is no (174, 199, 490164)-net in base 2, because
- 1 times m-reduction [i] would yield (174, 198, 490164)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 401740 883657 100137 741072 919558 496984 750511 819475 048772 674732 > 2198 [i]