Best Known (177, 177+25, s)-Nets in Base 2
(177, 177+25, 5464)-Net over F2 — Constructive and digital
Digital (177, 202, 5464)-net over F2, using
- 23 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(2199, 5464, F2, 25, 25) (dual of [(5464, 25), 136401, 26]-NRT-code), using
(177, 177+25, 10929)-Net over F2 — Digital
Digital (177, 202, 10929)-net over F2, using
- 22 times duplication [i] based on digital (175, 200, 10929)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2200, 10929, F2, 6, 25) (dual of [(10929, 6), 65374, 26]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2200, 65574, F2, 25) (dual of [65574, 65374, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2200, 65576, F2, 25) (dual of [65576, 65376, 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(27, 39, F2, 3) (dual of [39, 32, 4]-code or 39-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(2200, 65576, F2, 25) (dual of [65576, 65376, 26]-code), using
- OOA 6-folding [i] based on linear OA(2200, 65574, F2, 25) (dual of [65574, 65374, 26]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2200, 10929, F2, 6, 25) (dual of [(10929, 6), 65374, 26]-NRT-code), using
(177, 177+25, 582909)-Net in Base 2 — Upper bound on s
There is no (177, 202, 582910)-net in base 2, because
- 1 times m-reduction [i] would yield (177, 201, 582910)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 3 213938 492344 807835 990780 197338 969962 526888 473866 941119 773643 > 2201 [i]