Best Known (127−10, 127, s)-Nets in Base 8
(127−10, 127, large)-Net over F8 — Constructive and digital
Digital (117, 127, large)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 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 (112, 122, 8388602)-net over F8, using
- trace code for nets [i] based on digital (51, 61, 4194301)-net over F64, using
- net defined by OOA [i] based on linear OOA(6461, 4194301, F64, 15, 10) (dual of [(4194301, 15), 62914454, 11]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(6461, large, F64, 3, 10), using
- generalized (u, u+v)-construction [i] based on
- linear OOA(647, 2796201, F64, 3, 3) (dual of [(2796201, 3), 8388596, 4]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(647, 8388602, F64, 3, 3) (dual of [(8388602, 3), 25165799, 4]-NRT-code), using
- OOA stacking with additional row [i] based on linear OA(647, large, F64, 3) (dual of [large, large−7, 4]-code), using
- discarding factors / shortening the dual code based on linear OOA(647, 8388602, F64, 3, 3) (dual of [(8388602, 3), 25165799, 4]-NRT-code), using
- linear OOA(6417, 2796201, F64, 3, 5) (dual of [(2796201, 3), 8388586, 6]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6417, 4194301, F64, 3, 5) (dual of [(4194301, 3), 12582886, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(6417, large, F64, 5) (dual of [large, large−17, 6]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- OOA 2-folding and stacking with additional row [i] based on linear OA(6417, large, F64, 5) (dual of [large, large−17, 6]-code), using
- discarding factors / shortening the dual code based on linear OOA(6417, 4194301, F64, 3, 5) (dual of [(4194301, 3), 12582886, 6]-NRT-code), using
- linear OOA(6437, 2796201, F64, 3, 10) (dual of [(2796201, 3), 8388566, 11]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6437, large, F64, 10) (dual of [large, large−37, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- OOA 3-folding [i] based on linear OA(6437, large, F64, 10) (dual of [large, large−37, 11]-code), using
- linear OOA(647, 2796201, F64, 3, 3) (dual of [(2796201, 3), 8388596, 4]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OOA(6461, large, F64, 3, 10), using
- net defined by OOA [i] based on linear OOA(6461, 4194301, F64, 15, 10) (dual of [(4194301, 15), 62914454, 11]-NRT-code), using
- trace code for nets [i] based on digital (51, 61, 4194301)-net over F64, using
- digital (0, 5, 9)-net over F8, using
(127−10, 127, large)-Net in Base 8 — Upper bound on s
There is no (117, 127, large)-net in base 8, because
- 8 times m-reduction [i] would yield (117, 119, large)-net in base 8, but