Best Known (202−20, 202, s)-Nets in Base 2
(202−20, 202, 104859)-Net over F2 — Constructive and digital
Digital (182, 202, 104859)-net over F2, using
- t-expansion [i] based on digital (181, 202, 104859)-net over F2, using
- net defined by OOA [i] based on linear OOA(2202, 104859, F2, 21, 21) (dual of [(104859, 21), 2201837, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2202, 1048591, F2, 21) (dual of [1048591, 1048389, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2202, 1048597, F2, 21) (dual of [1048597, 1048395, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(2201, 1048576, F2, 21) (dual of [1048576, 1048375, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2181, 1048576, F2, 19) (dual of [1048576, 1048395, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(21, 21, F2, 1) (dual of [21, 20, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(2202, 1048597, F2, 21) (dual of [1048597, 1048395, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2202, 1048591, F2, 21) (dual of [1048591, 1048389, 22]-code), using
- net defined by OOA [i] based on linear OOA(2202, 104859, F2, 21, 21) (dual of [(104859, 21), 2201837, 22]-NRT-code), using
(202−20, 202, 174766)-Net over F2 — Digital
Digital (182, 202, 174766)-net over F2, using
- 21 times duplication [i] based on digital (181, 201, 174766)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2201, 174766, F2, 6, 20) (dual of [(174766, 6), 1048395, 21]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2201, 1048596, F2, 20) (dual of [1048596, 1048395, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(2201, 1048597, F2, 20) (dual of [1048597, 1048396, 21]-code), using
- 1 times truncation [i] based on linear OA(2202, 1048598, F2, 21) (dual of [1048598, 1048396, 22]-code), using
- construction X4 applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(2201, 1048576, F2, 21) (dual of [1048576, 1048375, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2181, 1048576, F2, 19) (dual of [1048576, 1048395, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(221, 22, F2, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,2)), using
- dual of repetition code with length 22 [i]
- linear OA(21, 22, F2, 1) (dual of [22, 21, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(20) ⊂ Ce(18) [i] based on
- 1 times truncation [i] based on linear OA(2202, 1048598, F2, 21) (dual of [1048598, 1048396, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2201, 1048597, F2, 20) (dual of [1048597, 1048396, 21]-code), using
- OOA 6-folding [i] based on linear OA(2201, 1048596, F2, 20) (dual of [1048596, 1048395, 21]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2201, 174766, F2, 6, 20) (dual of [(174766, 6), 1048395, 21]-NRT-code), using
(202−20, 202, 5454828)-Net in Base 2 — Upper bound on s
There is no (182, 202, 5454829)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 6 427763 935993 219899 679765 253711 250172 467571 804883 575435 889894 > 2202 [i]