Best Known (213−27, 213, s)-Nets in Base 2
(213−27, 213, 5042)-Net over F2 — Constructive and digital
Digital (186, 213, 5042)-net over F2, using
- 23 times duplication [i] based on digital (183, 210, 5042)-net over F2, using
- net defined by OOA [i] based on linear OOA(2210, 5042, F2, 27, 27) (dual of [(5042, 27), 135924, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2210, 65547, F2, 27) (dual of [65547, 65337, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2210, 65553, F2, 27) (dual of [65553, 65343, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- linear OA(2209, 65536, F2, 27) (dual of [65536, 65327, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2193, 65536, F2, 25) (dual of [65536, 65343, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(21, 17, F2, 1) (dual of [17, 16, 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(26) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(2210, 65553, F2, 27) (dual of [65553, 65343, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2210, 65547, F2, 27) (dual of [65547, 65337, 28]-code), using
- net defined by OOA [i] based on linear OOA(2210, 5042, F2, 27, 27) (dual of [(5042, 27), 135924, 28]-NRT-code), using
(213−27, 213, 10809)-Net over F2 — Digital
Digital (186, 213, 10809)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2213, 10809, F2, 6, 27) (dual of [(10809, 6), 64641, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2213, 10926, F2, 6, 27) (dual of [(10926, 6), 65343, 28]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2213, 65556, F2, 27) (dual of [65556, 65343, 28]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2210, 65553, F2, 27) (dual of [65553, 65343, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- linear OA(2209, 65536, F2, 27) (dual of [65536, 65327, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2193, 65536, F2, 25) (dual of [65536, 65343, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(21, 17, F2, 1) (dual of [17, 16, 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(26) ⊂ Ce(24) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(2210, 65553, F2, 27) (dual of [65553, 65343, 28]-code), using
- OOA 6-folding [i] based on linear OA(2213, 65556, F2, 27) (dual of [65556, 65343, 28]-code), using
- discarding factors / shortening the dual code based on linear OOA(2213, 10926, F2, 6, 27) (dual of [(10926, 6), 65343, 28]-NRT-code), using
(213−27, 213, 459719)-Net in Base 2 — Upper bound on s
There is no (186, 213, 459720)-net in base 2, because
- 1 times m-reduction [i] would yield (186, 212, 459720)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 6582 141102 753351 680568 433095 800166 295839 382633 003931 073324 154824 > 2212 [i]