Best Known (163−20, 163, s)-Nets in Base 2
(163−20, 163, 6555)-Net over F2 — Constructive and digital
Digital (143, 163, 6555)-net over F2, using
- 21 times duplication [i] based on digital (142, 162, 6555)-net over F2, using
- t-expansion [i] based on digital (141, 162, 6555)-net over F2, using
- net defined by OOA [i] based on linear OOA(2162, 6555, F2, 21, 21) (dual of [(6555, 21), 137493, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2162, 65551, F2, 21) (dual of [65551, 65389, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2162, 65553, F2, 21) (dual of [65553, 65391, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(2161, 65536, F2, 21) (dual of [65536, 65375, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2145, 65536, F2, 19) (dual of [65536, 65391, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(20) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(2162, 65553, F2, 21) (dual of [65553, 65391, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2162, 65551, F2, 21) (dual of [65551, 65389, 22]-code), using
- net defined by OOA [i] based on linear OOA(2162, 6555, F2, 21, 21) (dual of [(6555, 21), 137493, 22]-NRT-code), using
- t-expansion [i] based on digital (141, 162, 6555)-net over F2, using
(163−20, 163, 13110)-Net over F2 — Digital
Digital (143, 163, 13110)-net over F2, using
- 22 times duplication [i] based on digital (141, 161, 13110)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2161, 13110, F2, 5, 20) (dual of [(13110, 5), 65389, 21]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2161, 65550, F2, 20) (dual of [65550, 65389, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(2161, 65552, F2, 20) (dual of [65552, 65391, 21]-code), using
- 1 times truncation [i] based on linear OA(2162, 65553, F2, 21) (dual of [65553, 65391, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(2161, 65536, F2, 21) (dual of [65536, 65375, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2145, 65536, F2, 19) (dual of [65536, 65391, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(20) ⊂ Ce(18) [i] based on
- 1 times truncation [i] based on linear OA(2162, 65553, F2, 21) (dual of [65553, 65391, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2161, 65552, F2, 20) (dual of [65552, 65391, 21]-code), using
- OOA 5-folding [i] based on linear OA(2161, 65550, F2, 20) (dual of [65550, 65389, 21]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2161, 13110, F2, 5, 20) (dual of [(13110, 5), 65389, 21]-NRT-code), using
(163−20, 163, 365382)-Net in Base 2 — Upper bound on s
There is no (143, 163, 365383)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 11 692104 290371 555130 816888 987284 889200 215341 154008 > 2163 [i]