Best Known (63, 71, s)-Nets in Base 4
(63, 71, 1048586)-Net over F4 — Constructive and digital
Digital (63, 71, 1048586)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 10)-net over F4, using
- 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
- net defined by OOA [i] based on linear OOA(466, 1048576, F4, 8, 8) (dual of [(1048576, 8), 8388542, 9]-NRT-code), using
(63, 71, 4194341)-Net over F4 — Digital
Digital (63, 71, 4194341)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(471, 4194341, F4, 8) (dual of [4194341, 4194270, 9]-code), using
- 1 times truncation [i] based on linear OA(472, 4194342, F4, 9) (dual of [4194342, 4194270, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(4) [i] based on
- linear OA(467, 4194304, F4, 9) (dual of [4194304, 4194237, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(434, 4194304, F4, 5) (dual of [4194304, 4194270, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(45, 38, F4, 3) (dual of [38, 33, 4]-code or 38-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(8) ⊂ Ce(4) [i] based on
- 1 times truncation [i] based on linear OA(472, 4194342, F4, 9) (dual of [4194342, 4194270, 10]-code), using
(63, 71, large)-Net in Base 4 — Upper bound on s
There is no (63, 71, large)-net in base 4, because
- 6 times m-reduction [i] would yield (63, 65, large)-net in base 4, but