Best Known (77−9, 77, s)-Nets in Base 4
(77−9, 77, 2097155)-Net over F4 — Constructive and digital
Digital (68, 77, 2097155)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 5)-net over F4, using
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 0 and N(F) ≥ 5, using
- the rational function field F4(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- digital (64, 73, 2097150)-net over F4, using
- net defined by OOA [i] based on linear OOA(473, 2097150, F4, 9, 9) (dual of [(2097150, 9), 18874277, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(473, 8388601, F4, 9) (dual of [8388601, 8388528, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(473, large, F4, 9) (dual of [large, large−73, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(473, large, F4, 9) (dual of [large, large−73, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(473, 8388601, F4, 9) (dual of [8388601, 8388528, 10]-code), using
- net defined by OOA [i] based on linear OOA(473, 2097150, F4, 9, 9) (dual of [(2097150, 9), 18874277, 10]-NRT-code), using
- digital (0, 4, 5)-net over F4, using
(77−9, 77, 4194306)-Net over F4 — Digital
Digital (68, 77, 4194306)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(477, 4194306, F4, 2, 9) (dual of [(4194306, 2), 8388535, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(44, 5, F4, 2, 4) (dual of [(5, 2), 6, 5]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(2;6,4) [i]
- linear OOA(473, 4194301, F4, 2, 9) (dual of [(4194301, 2), 8388529, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(473, 8388602, F4, 9) (dual of [8388602, 8388529, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(473, large, F4, 9) (dual of [large, large−73, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(473, large, F4, 9) (dual of [large, large−73, 10]-code), using
- OOA 2-folding [i] based on linear OA(473, 8388602, F4, 9) (dual of [8388602, 8388529, 10]-code), using
- linear OOA(44, 5, F4, 2, 4) (dual of [(5, 2), 6, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
(77−9, 77, large)-Net in Base 4 — Upper bound on s
There is no (68, 77, large)-net in base 4, because
- 7 times m-reduction [i] would yield (68, 70, large)-net in base 4, but