Best Known (172−19, 172, s)-Nets in Base 8
(172−19, 172, 1864393)-Net over F8 — Constructive and digital
Digital (153, 172, 1864393)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (17, 26, 260)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (4, 8, 130)-net over F8, using
- trace code for nets [i] based on digital (0, 4, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- trace code for nets [i] based on digital (0, 4, 65)-net over F64, using
- digital (9, 18, 130)-net over F8, using
- trace code for nets [i] based on digital (0, 9, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- trace code for nets [i] based on digital (0, 9, 65)-net over F64, using
- digital (4, 8, 130)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (127, 146, 1864133)-net over F8, using
- net defined by OOA [i] based on linear OOA(8146, 1864133, F8, 21, 19) (dual of [(1864133, 21), 39146647, 20]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(8146, 5592400, F8, 3, 19) (dual of [(5592400, 3), 16777054, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8146, 5592402, F8, 3, 19) (dual of [(5592402, 3), 16777060, 20]-NRT-code), using
- trace code [i] based on linear OOA(6473, 2796201, F64, 3, 19) (dual of [(2796201, 3), 8388530, 20]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6473, large, F64, 19) (dual of [large, large−73, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- OOA 3-folding [i] based on linear OA(6473, large, F64, 19) (dual of [large, large−73, 20]-code), using
- trace code [i] based on linear OOA(6473, 2796201, F64, 3, 19) (dual of [(2796201, 3), 8388530, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8146, 5592402, F8, 3, 19) (dual of [(5592402, 3), 16777060, 20]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(8146, 5592400, F8, 3, 19) (dual of [(5592400, 3), 16777054, 20]-NRT-code), using
- net defined by OOA [i] based on linear OOA(8146, 1864133, F8, 21, 19) (dual of [(1864133, 21), 39146647, 20]-NRT-code), using
- digital (17, 26, 260)-net over F8, using
(172−19, 172, 1864647)-Net in Base 8 — Constructive
(153, 172, 1864647)-net in base 8, using
- 82 times duplication [i] based on (151, 170, 1864647)-net in base 8, using
- (u, u+v)-construction [i] based on
- (15, 24, 514)-net in base 8, using
- base change [i] based on digital (9, 18, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 9, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 9, 257)-net over F256, using
- base change [i] based on digital (9, 18, 514)-net over F16, using
- digital (127, 146, 1864133)-net over F8, using
- net defined by OOA [i] based on linear OOA(8146, 1864133, F8, 21, 19) (dual of [(1864133, 21), 39146647, 20]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(8146, 5592400, F8, 3, 19) (dual of [(5592400, 3), 16777054, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8146, 5592402, F8, 3, 19) (dual of [(5592402, 3), 16777060, 20]-NRT-code), using
- trace code [i] based on linear OOA(6473, 2796201, F64, 3, 19) (dual of [(2796201, 3), 8388530, 20]-NRT-code), using
- OOA 3-folding [i] based on linear OA(6473, large, F64, 19) (dual of [large, large−73, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- OOA 3-folding [i] based on linear OA(6473, large, F64, 19) (dual of [large, large−73, 20]-code), using
- trace code [i] based on linear OOA(6473, 2796201, F64, 3, 19) (dual of [(2796201, 3), 8388530, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8146, 5592402, F8, 3, 19) (dual of [(5592402, 3), 16777060, 20]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(8146, 5592400, F8, 3, 19) (dual of [(5592400, 3), 16777054, 20]-NRT-code), using
- net defined by OOA [i] based on linear OOA(8146, 1864133, F8, 21, 19) (dual of [(1864133, 21), 39146647, 20]-NRT-code), using
- (15, 24, 514)-net in base 8, using
- (u, u+v)-construction [i] based on
(172−19, 172, large)-Net over F8 — Digital
Digital (153, 172, large)-net over F8, using
- 84 times duplication [i] based on digital (149, 168, large)-net over F8, using
- t-expansion [i] based on digital (144, 168, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8168, large, F8, 24) (dual of [large, large−168, 25]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8168, large, F8, 24) (dual of [large, large−168, 25]-code), using
- t-expansion [i] based on digital (144, 168, large)-net over F8, using
(172−19, 172, large)-Net in Base 8 — Upper bound on s
There is no (153, 172, large)-net in base 8, because
- 17 times m-reduction [i] would yield (153, 155, large)-net in base 8, but