Best Known (199−20, 199, s)-Nets in Base 2
(199−20, 199, 52433)-Net over F2 — Constructive and digital
Digital (179, 199, 52433)-net over F2, using
- 21 times duplication [i] based on digital (178, 198, 52433)-net over F2, using
- t-expansion [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
- t-expansion [i] based on digital (177, 198, 52433)-net over F2, using
(199−20, 199, 89697)-Net over F2 — Digital
Digital (179, 199, 89697)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2199, 89697, F2, 5, 20) (dual of [(89697, 5), 448286, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2199, 104867, F2, 5, 20) (dual of [(104867, 5), 524136, 21]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2199, 524335, F2, 20) (dual of [524335, 524136, 21]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2197, 524333, F2, 20) (dual of [524333, 524136, 21]-code), using
- 1 times truncation [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
- 1 times truncation [i] based on linear OA(2198, 524334, F2, 21) (dual of [524334, 524136, 22]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2197, 524333, F2, 20) (dual of [524333, 524136, 21]-code), using
- OOA 5-folding [i] based on linear OA(2199, 524335, F2, 20) (dual of [524335, 524136, 21]-code), using
- discarding factors / shortening the dual code based on linear OOA(2199, 104867, F2, 5, 20) (dual of [(104867, 5), 524136, 21]-NRT-code), using
(199−20, 199, 4430694)-Net in Base 2 — Upper bound on s
There is no (179, 199, 4430695)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 803470 120919 764146 785535 354805 457908 515556 278087 904845 330884 > 2199 [i]