Best Known (57, 57+8, s)-Nets in Base 4
(57, 57+8, 262154)-Net over F4 — Constructive and digital
Digital (57, 65, 262154)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 10)-net over F4, using
- digital (52, 60, 262144)-net over F4, using
- net defined by OOA [i] based on linear OOA(460, 262144, F4, 8, 8) (dual of [(262144, 8), 2097092, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(460, 1048576, F4, 8) (dual of [1048576, 1048516, 9]-code), using
- 1 times truncation [i] based on linear OA(461, 1048577, F4, 9) (dual of [1048577, 1048516, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1048577 | 420−1, 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(461, 1048577, F4, 9) (dual of [1048577, 1048516, 10]-code), using
- OA 4-folding and stacking [i] based on linear OA(460, 1048576, F4, 8) (dual of [1048576, 1048516, 9]-code), using
- net defined by OOA [i] based on linear OOA(460, 262144, F4, 8, 8) (dual of [(262144, 8), 2097092, 9]-NRT-code), using
(57, 57+8, 1048610)-Net over F4 — Digital
Digital (57, 65, 1048610)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(465, 1048610, F4, 8) (dual of [1048610, 1048545, 9]-code), using
- 1 times truncation [i] based on linear OA(466, 1048611, F4, 9) (dual of [1048611, 1048545, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(4) [i] based on
- 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(431, 1048576, F4, 5) (dual of [1048576, 1048545, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(8) ⊂ Ce(4) [i] based on
- 1 times truncation [i] based on linear OA(466, 1048611, F4, 9) (dual of [1048611, 1048545, 10]-code), using
(57, 57+8, large)-Net in Base 4 — Upper bound on s
There is no (57, 65, large)-net in base 4, because
- 6 times m-reduction [i] would yield (57, 59, large)-net in base 4, but