Best Known (84, 84+17, s)-Nets in Base 4
(84, 84+17, 8194)-Net over F4 — Constructive and digital
Digital (84, 101, 8194)-net over F4, using
- 41 times duplication [i] based on digital (83, 100, 8194)-net over F4, using
- net defined by OOA [i] based on linear OOA(4100, 8194, F4, 17, 17) (dual of [(8194, 17), 139198, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(4100, 65553, F4, 17) (dual of [65553, 65453, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(4100, 65555, F4, 17) (dual of [65555, 65455, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(497, 65536, F4, 17) (dual of [65536, 65439, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(481, 65536, F4, 14) (dual of [65536, 65455, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(43, 19, F4, 2) (dual of [19, 16, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(4100, 65555, F4, 17) (dual of [65555, 65455, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(4100, 65553, F4, 17) (dual of [65553, 65453, 18]-code), using
- net defined by OOA [i] based on linear OOA(4100, 8194, F4, 17, 17) (dual of [(8194, 17), 139198, 18]-NRT-code), using
(84, 84+17, 32778)-Net over F4 — Digital
Digital (84, 101, 32778)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4101, 32778, F4, 2, 17) (dual of [(32778, 2), 65455, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4101, 65556, F4, 17) (dual of [65556, 65455, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(4101, 65557, F4, 17) (dual of [65557, 65456, 18]-code), using
- construction XX applied to Ce(16) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- linear OA(497, 65536, F4, 17) (dual of [65536, 65439, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(481, 65536, F4, 14) (dual of [65536, 65455, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(473, 65536, F4, 13) (dual of [65536, 65463, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(43, 20, F4, 2) (dual of [20, 17, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(16) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(4101, 65557, F4, 17) (dual of [65557, 65456, 18]-code), using
- OOA 2-folding [i] based on linear OA(4101, 65556, F4, 17) (dual of [65556, 65455, 18]-code), using
(84, 84+17, large)-Net in Base 4 — Upper bound on s
There is no (84, 101, large)-net in base 4, because
- 15 times m-reduction [i] would yield (84, 86, large)-net in base 4, but