Best Known (198−24, 198, s)-Nets in Base 2
(198−24, 198, 5464)-Net over F2 — Constructive and digital
Digital (174, 198, 5464)-net over F2, using
- net defined by OOA [i] based on linear OOA(2198, 5464, F2, 24, 24) (dual of [(5464, 24), 130938, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(2198, 65568, F2, 24) (dual of [65568, 65370, 25]-code), using
- 1 times truncation [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
- 1 times truncation [i] based on linear OA(2199, 65569, F2, 25) (dual of [65569, 65370, 26]-code), using
- OA 12-folding and stacking [i] based on linear OA(2198, 65568, F2, 24) (dual of [65568, 65370, 25]-code), using
(198−24, 198, 12734)-Net over F2 — Digital
Digital (174, 198, 12734)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2198, 12734, F2, 5, 24) (dual of [(12734, 5), 63472, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2198, 13113, F2, 5, 24) (dual of [(13113, 5), 65367, 25]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2198, 65565, F2, 24) (dual of [65565, 65367, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(2198, 65568, F2, 24) (dual of [65568, 65370, 25]-code), using
- 1 times truncation [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
- 1 times truncation [i] based on linear OA(2199, 65569, F2, 25) (dual of [65569, 65370, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2198, 65568, F2, 24) (dual of [65568, 65370, 25]-code), using
- OOA 5-folding [i] based on linear OA(2198, 65565, F2, 24) (dual of [65565, 65367, 25]-code), using
- discarding factors / shortening the dual code based on linear OOA(2198, 13113, F2, 5, 24) (dual of [(13113, 5), 65367, 25]-NRT-code), using
(198−24, 198, 490163)-Net in Base 2 — Upper bound on s
There is no (174, 198, 490164)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 401740 883657 100137 741072 919558 496984 750511 819475 048772 674732 > 2198 [i]