Best Known (68, 68+9, s)-Nets in Base 27
(68, 68+9, large)-Net over F27 — Constructive and digital
Digital (68, 77, large)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (12, 16, 4194301)-net over F27, using
- net defined by OOA [i] based on linear OOA(2716, 4194301, F27, 4, 4) (dual of [(4194301, 4), 16777188, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(2716, 8388602, F27, 4) (dual of [8388602, 8388586, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(2716, large, F27, 4) (dual of [large, large−16, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(2716, large, F27, 4) (dual of [large, large−16, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(2716, 8388602, F27, 4) (dual of [8388602, 8388586, 5]-code), using
- net defined by OOA [i] based on linear OOA(2716, 4194301, F27, 4, 4) (dual of [(4194301, 4), 16777188, 5]-NRT-code), using
- digital (52, 61, 4194356)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (12, 16, 4194301)-net over F27 (see above)
- digital (36, 45, 2097178)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 0 and N(F) ≥ 28, using
- the rational function field F27(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- 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
- digital (0, 4, 28)-net over F27, using
- (u, u+v)-construction [i] based on
- (u, u+v)-construction [i] based on
- digital (12, 16, 4194301)-net over F27, using
(68, 68+9, large)-Net in Base 27 — Upper bound on s
There is no (68, 77, large)-net in base 27, because
- 7 times m-reduction [i] would yield (68, 70, large)-net in base 27, but