Best Known (51−8, 51, s)-Nets in Base 4
(51−8, 51, 16388)-Net over F4 — Constructive and digital
Digital (43, 51, 16388)-net over F4, using
- 1 times m-reduction [i] based on digital (43, 52, 16388)-net over F4, using
- net defined by OOA [i] based on linear OOA(452, 16388, F4, 9, 9) (dual of [(16388, 9), 147440, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(452, 65553, F4, 9) (dual of [65553, 65501, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(452, 65555, F4, 9) (dual of [65555, 65503, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(449, 65536, F4, 9) (dual of [65536, 65487, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(433, 65536, F4, 6) (dual of [65536, 65503, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(8) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(452, 65555, F4, 9) (dual of [65555, 65503, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(452, 65553, F4, 9) (dual of [65553, 65501, 10]-code), using
- net defined by OOA [i] based on linear OOA(452, 16388, F4, 9, 9) (dual of [(16388, 9), 147440, 10]-NRT-code), using
(51−8, 51, 65556)-Net over F4 — Digital
Digital (43, 51, 65556)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(451, 65556, F4, 8) (dual of [65556, 65505, 9]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(450, 65554, F4, 8) (dual of [65554, 65504, 9]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(449, 65536, F4, 9) (dual of [65536, 65487, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(433, 65536, F4, 6) (dual of [65536, 65503, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(417, 18, F4, 17) (dual of [18, 1, 18]-code or 18-arc in PG(16,4)), using
- dual of repetition code with length 18 [i]
- linear OA(41, 18, F4, 1) (dual of [18, 17, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(450, 65555, F4, 7) (dual of [65555, 65505, 8]-code), using Gilbert–Varšamov bound and bm = 450 > Vbs−1(k−1) = 80334 868316 187991 022541 334696 [i]
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(450, 65554, F4, 8) (dual of [65554, 65504, 9]-code), using
- construction X with Varšamov bound [i] based on
(51−8, 51, large)-Net in Base 4 — Upper bound on s
There is no (43, 51, large)-net in base 4, because
- 6 times m-reduction [i] would yield (43, 45, large)-net in base 4, but