Best Known (54, 54+9, s)-Nets in Base 9
(54, 54+9, 1195823)-Net over F9 — Constructive and digital
Digital (54, 63, 1195823)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (3, 7, 82)-net over F9, using
- base reduction for projective spaces (embedding PG(3,81) in PG(6,9)) for nets [i] based on digital (0, 4, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- base reduction for projective spaces (embedding PG(3,81) in PG(6,9)) for nets [i] based on digital (0, 4, 82)-net over F81, using
- digital (47, 56, 1195741)-net over F9, using
- net defined by OOA [i] based on linear OOA(956, 1195741, F9, 9, 9) (dual of [(1195741, 9), 10761613, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(956, 4782965, F9, 9) (dual of [4782965, 4782909, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(956, 4782968, F9, 9) (dual of [4782968, 4782912, 10]-code), using
- 1 times truncation [i] based on linear OA(957, 4782969, F9, 10) (dual of [4782969, 4782912, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- 1 times truncation [i] based on linear OA(957, 4782969, F9, 10) (dual of [4782969, 4782912, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(956, 4782968, F9, 9) (dual of [4782968, 4782912, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(956, 4782965, F9, 9) (dual of [4782965, 4782909, 10]-code), using
- net defined by OOA [i] based on linear OOA(956, 1195741, F9, 9, 9) (dual of [(1195741, 9), 10761613, 10]-NRT-code), using
- digital (3, 7, 82)-net over F9, using
(54, 54+9, 2097150)-Net in Base 9 — Constructive
(54, 63, 2097150)-net in base 9, using
- base change [i] based on digital (33, 42, 2097150)-net over F27, using
- 271 times duplication [i] based on digital (32, 41, 2097150)-net over F27, using
- net defined by OOA [i] based on linear OOA(2741, 2097150, F27, 9, 9) (dual of [(2097150, 9), 18874309, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2741, 8388601, F27, 9) (dual of [8388601, 8388560, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(2741, large, F27, 9) (dual of [large, large−41, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, 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(2741, large, F27, 9) (dual of [large, large−41, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2741, 8388601, F27, 9) (dual of [8388601, 8388560, 10]-code), using
- net defined by OOA [i] based on linear OOA(2741, 2097150, F27, 9, 9) (dual of [(2097150, 9), 18874309, 10]-NRT-code), using
- 271 times duplication [i] based on digital (32, 41, 2097150)-net over F27, using
(54, 54+9, large)-Net over F9 — Digital
Digital (54, 63, large)-net over F9, using
- 92 times duplication [i] based on digital (52, 61, large)-net over F9, using
(54, 54+9, large)-Net in Base 9 — Upper bound on s
There is no (54, 63, large)-net in base 9, because
- 7 times m-reduction [i] would yield (54, 56, large)-net in base 9, but