Best Known (168−8, 168, s)-Nets in Base 8
(168−8, 168, large)-Net over F8 — Constructive and digital
Digital (160, 168, large)-net over F8, using
- t-expansion [i] based on digital (159, 168, large)-net over F8, using
- 4 times m-reduction [i] based on digital (159, 172, large)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 9)-net over F8, using
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 0 and N(F) ≥ 9, using
- the rational function field F8(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- digital (153, 166, 8388602)-net over F8, using
- trace code for nets [i] based on digital (70, 83, 4194301)-net over F64, using
- net defined by OOA [i] based on linear OOA(6483, 4194301, F64, 15, 13) (dual of [(4194301, 15), 62914432, 14]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(6483, large, F64, 3, 13), using
- generalized (u, u+v)-construction [i] based on
- linear OOA(6413, 2796201, F64, 3, 4) (dual of [(2796201, 3), 8388590, 5]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6413, 4194301, F64, 3, 4) (dual of [(4194301, 3), 12582890, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(6413, 8388602, F64, 4) (dual of [8388602, 8388589, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(6413, large, F64, 4) (dual of [large, large−13, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(6413, large, F64, 4) (dual of [large, large−13, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(6413, 8388602, F64, 4) (dual of [8388602, 8388589, 5]-code), using
- discarding factors / shortening the dual code based on linear OOA(6413, 4194301, F64, 3, 4) (dual of [(4194301, 3), 12582890, 5]-NRT-code), using
- linear OOA(6421, 2796201, F64, 3, 6) (dual of [(2796201, 3), 8388582, 7]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6421, large, F64, 6) (dual of [large, large−21, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OOA 3-folding [i] based on linear OA(6421, large, F64, 6) (dual of [large, large−21, 7]-code), using
- linear OOA(6449, 2796201, F64, 3, 13) (dual of [(2796201, 3), 8388554, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6449, large, F64, 13) (dual of [large, large−49, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- OOA 3-folding [i] based on linear OA(6449, large, F64, 13) (dual of [large, large−49, 14]-code), using
- linear OOA(6413, 2796201, F64, 3, 4) (dual of [(2796201, 3), 8388590, 5]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OOA(6483, large, F64, 3, 13), using
- net defined by OOA [i] based on linear OOA(6483, 4194301, F64, 15, 13) (dual of [(4194301, 15), 62914432, 14]-NRT-code), using
- trace code for nets [i] based on digital (70, 83, 4194301)-net over F64, using
- digital (0, 6, 9)-net over F8, using
- (u, u+v)-construction [i] based on
- 4 times m-reduction [i] based on digital (159, 172, large)-net over F8, using
(168−8, 168, large)-Net in Base 8 — Upper bound on s
There is no (160, 168, large)-net in base 8, because
- 6 times m-reduction [i] would yield (160, 162, large)-net in base 8, but