Best Known (199−21, 199, s)-Nets in Base 2
(199−21, 199, 52433)-Net over F2 — Constructive and digital
Digital (178, 199, 52433)-net over F2, using
- 21 times duplication [i] based on digital (177, 198, 52433)-net over F2, using
- net defined by OOA [i] based on linear OOA(2198, 52433, F2, 21, 21) (dual of [(52433, 21), 1100895, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2198, 524331, F2, 21) (dual of [524331, 524133, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2198, 524334, F2, 21) (dual of [524334, 524136, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(2191, 524289, F2, 21) (dual of [524289, 524098, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 524289 | 238−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2153, 524289, F2, 17) (dual of [524289, 524136, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 524289 | 238−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(27, 45, F2, 3) (dual of [45, 38, 4]-code or 45-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([0,10]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2198, 524334, F2, 21) (dual of [524334, 524136, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2198, 524331, F2, 21) (dual of [524331, 524133, 22]-code), using
- net defined by OOA [i] based on linear OOA(2198, 52433, F2, 21, 21) (dual of [(52433, 21), 1100895, 22]-NRT-code), using
(199−21, 199, 85363)-Net over F2 — Digital
Digital (178, 199, 85363)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2199, 85363, F2, 6, 21) (dual of [(85363, 6), 511979, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2199, 87389, F2, 6, 21) (dual of [(87389, 6), 524135, 22]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2198, 87389, F2, 6, 21) (dual of [(87389, 6), 524136, 22]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2198, 524334, F2, 21) (dual of [524334, 524136, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(2191, 524289, F2, 21) (dual of [524289, 524098, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 524289 | 238−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2153, 524289, F2, 17) (dual of [524289, 524136, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 524289 | 238−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(27, 45, F2, 3) (dual of [45, 38, 4]-code or 45-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([0,10]) ⊂ C([0,8]) [i] based on
- OOA 6-folding [i] based on linear OA(2198, 524334, F2, 21) (dual of [524334, 524136, 22]-code), using
- 21 times duplication [i] based on linear OOA(2198, 87389, F2, 6, 21) (dual of [(87389, 6), 524136, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2199, 87389, F2, 6, 21) (dual of [(87389, 6), 524135, 22]-NRT-code), using
(199−21, 199, 4133983)-Net in Base 2 — Upper bound on s
There is no (178, 199, 4133984)-net in base 2, because
- 1 times m-reduction [i] would yield (178, 198, 4133984)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 401735 410867 121857 677877 758625 695574 233251 750338 772031 416493 > 2198 [i]