Best Known (122, 122+15, s)-Nets in Base 8
(122, 122+15, 2397148)-Net over F8 — Constructive and digital
Digital (122, 137, 2397148)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (16, 23, 406)-net over F8, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 1, 73)-net over F8, using
- s-reduction based on digital (0, 1, s)-net over F8 with arbitrarily large s, using
- digital (1, 3, 73)-net over F8, using
- digital (2, 5, 130)-net over F8, using
- s-reduction based on digital (2, 5, 208)-net over F8, using
- net defined by OOA [i] based on linear OOA(85, 208, F8, 3, 3) (dual of [(208, 3), 619, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(85, 208, F8, 2, 3) (dual of [(208, 2), 411, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(85, 208, F8, 3, 3) (dual of [(208, 3), 619, 4]-NRT-code), using
- s-reduction based on digital (2, 5, 208)-net over F8, using
- digital (7, 14, 130)-net over F8, using
- trace code for nets [i] based on digital (0, 7, 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, 7, 65)-net over F64, using
- digital (0, 1, 73)-net over F8, using
- generalized (u, u+v)-construction [i] based on
- digital (99, 114, 2396742)-net over F8, using
- trace code for nets [i] based on digital (42, 57, 1198371)-net over F64, using
- net defined by OOA [i] based on linear OOA(6457, 1198371, F64, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6457, 8388598, F64, 15) (dual of [8388598, 8388541, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6457, 8388598, F64, 15) (dual of [8388598, 8388541, 16]-code), using
- net defined by OOA [i] based on linear OOA(6457, 1198371, F64, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- trace code for nets [i] based on digital (42, 57, 1198371)-net over F64, using
- digital (16, 23, 406)-net over F8, using
(122, 122+15, 2397265)-Net in Base 8 — Constructive
(122, 137, 2397265)-net in base 8, using
- (u, u+v)-construction [i] based on
- (16, 23, 523)-net in base 8, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 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
- (13, 20, 514)-net in base 8, using
- base change [i] based on digital (8, 15, 514)-net over F16, using
- 1 times m-reduction [i] based on digital (8, 16, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 8, 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, 8, 257)-net over F256, using
- 1 times m-reduction [i] based on digital (8, 16, 514)-net over F16, using
- base change [i] based on digital (8, 15, 514)-net over F16, using
- digital (0, 3, 9)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (99, 114, 2396742)-net over F8, using
- trace code for nets [i] based on digital (42, 57, 1198371)-net over F64, using
- net defined by OOA [i] based on linear OOA(6457, 1198371, F64, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6457, 8388598, F64, 15) (dual of [8388598, 8388541, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6457, 8388598, F64, 15) (dual of [8388598, 8388541, 16]-code), using
- net defined by OOA [i] based on linear OOA(6457, 1198371, F64, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- trace code for nets [i] based on digital (42, 57, 1198371)-net over F64, using
- (16, 23, 523)-net in base 8, using
(122, 122+15, large)-Net over F8 — Digital
Digital (122, 137, large)-net over F8, using
- t-expansion [i] based on digital (119, 137, large)-net over F8, using
- 2 times m-reduction [i] based on digital (119, 139, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8139, large, F8, 20) (dual of [large, large−139, 21]-code), using
- 2 times code embedding in larger space [i] based on linear OA(8137, large, F8, 20) (dual of [large, large−137, 21]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 2 times code embedding in larger space [i] based on linear OA(8137, large, F8, 20) (dual of [large, large−137, 21]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8139, large, F8, 20) (dual of [large, large−139, 21]-code), using
- 2 times m-reduction [i] based on digital (119, 139, large)-net over F8, using
(122, 122+15, large)-Net in Base 8 — Upper bound on s
There is no (122, 137, large)-net in base 8, because
- 13 times m-reduction [i] would yield (122, 124, large)-net in base 8, but