Best Known (86, 99, s)-Nets in Base 4
(86, 99, 174774)-Net over F4 — Constructive and digital
Digital (86, 99, 174774)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 8, 12)-net over F4, using
- digital (78, 91, 174762)-net over F4, using
- net defined by OOA [i] based on linear OOA(491, 174762, F4, 13, 13) (dual of [(174762, 13), 2271815, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(491, 1048573, F4, 13) (dual of [1048573, 1048482, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(491, 1048576, F4, 13) (dual of [1048576, 1048485, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(491, 1048576, F4, 13) (dual of [1048576, 1048485, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(491, 1048573, F4, 13) (dual of [1048573, 1048482, 14]-code), using
- net defined by OOA [i] based on linear OOA(491, 174762, F4, 13, 13) (dual of [(174762, 13), 2271815, 14]-NRT-code), using
(86, 99, 524307)-Net over F4 — Digital
Digital (86, 99, 524307)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(499, 524307, F4, 2, 13) (dual of [(524307, 2), 1048515, 14]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(497, 524306, F4, 2, 13) (dual of [(524306, 2), 1048515, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(497, 1048612, F4, 13) (dual of [1048612, 1048515, 14]-code), using
- 1 times code embedding in larger space [i] based on linear OA(496, 1048611, F4, 13) (dual of [1048611, 1048515, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- linear OA(491, 1048576, F4, 13) (dual of [1048576, 1048485, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(461, 1048576, F4, 9) (dual of [1048576, 1048515, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(45, 35, F4, 3) (dual of [35, 30, 4]-code or 35-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(496, 1048611, F4, 13) (dual of [1048611, 1048515, 14]-code), using
- OOA 2-folding [i] based on linear OA(497, 1048612, F4, 13) (dual of [1048612, 1048515, 14]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(497, 524306, F4, 2, 13) (dual of [(524306, 2), 1048515, 14]-NRT-code), using
(86, 99, large)-Net in Base 4 — Upper bound on s
There is no (86, 99, large)-net in base 4, because
- 11 times m-reduction [i] would yield (86, 88, large)-net in base 4, but