Best Known (58, 58+8, s)-Nets in Base 4
(58, 58+8, 1048576)-Net over F4 — Constructive and digital
Digital (58, 66, 1048576)-net over F4, using
- net defined by OOA [i] based on linear OOA(466, 1048576, F4, 8, 8) (dual of [(1048576, 8), 8388542, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(466, 4194304, F4, 8) (dual of [4194304, 4194238, 9]-code), using
- 1 times truncation [i] based on linear OA(467, 4194305, F4, 9) (dual of [4194305, 4194238, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(467, 4194305, F4, 9) (dual of [4194305, 4194238, 10]-code), using
- OA 4-folding and stacking [i] based on linear OA(466, 4194304, F4, 8) (dual of [4194304, 4194238, 9]-code), using
(58, 58+8, 3322132)-Net over F4 — Digital
Digital (58, 66, 3322132)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(466, 3322132, F4, 8) (dual of [3322132, 3322066, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(466, 4194304, F4, 8) (dual of [4194304, 4194238, 9]-code), using
- 1 times truncation [i] based on linear OA(467, 4194305, F4, 9) (dual of [4194305, 4194238, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(467, 4194305, F4, 9) (dual of [4194305, 4194238, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(466, 4194304, F4, 8) (dual of [4194304, 4194238, 9]-code), using
(58, 58+8, large)-Net in Base 4 — Upper bound on s
There is no (58, 66, large)-net in base 4, because
- 6 times m-reduction [i] would yield (58, 60, large)-net in base 4, but