Best Known (136, 136+17, s)-Nets in Base 8
(136, 136+17, 2097358)-Net over F8 — Constructive and digital
Digital (136, 153, 2097358)-net over F8, using
- 81 times duplication [i] based on digital (135, 152, 2097358)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (14, 22, 208)-net over F8, using
- trace code for nets [i] based on digital (3, 11, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- trace code for nets [i] based on digital (3, 11, 104)-net over F64, using
- digital (113, 130, 2097150)-net over F8, using
- net defined by OOA [i] based on linear OOA(8130, 2097150, F8, 18, 17) (dual of [(2097150, 18), 37748570, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(8130, 8388601, F8, 2, 17) (dual of [(8388601, 2), 16777072, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8130, 8388602, F8, 2, 17) (dual of [(8388602, 2), 16777074, 18]-NRT-code), using
- trace code [i] based on linear OOA(6465, 4194301, F64, 2, 17) (dual of [(4194301, 2), 8388537, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6465, 8388602, F64, 17) (dual of [8388602, 8388537, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(6465, large, F64, 17) (dual of [large, large−65, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6465, large, F64, 17) (dual of [large, large−65, 18]-code), using
- OOA 2-folding [i] based on linear OA(6465, 8388602, F64, 17) (dual of [8388602, 8388537, 18]-code), using
- trace code [i] based on linear OOA(6465, 4194301, F64, 2, 17) (dual of [(4194301, 2), 8388537, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8130, 8388602, F8, 2, 17) (dual of [(8388602, 2), 16777074, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(8130, 8388601, F8, 2, 17) (dual of [(8388601, 2), 16777072, 18]-NRT-code), using
- net defined by OOA [i] based on linear OOA(8130, 2097150, F8, 18, 17) (dual of [(2097150, 18), 37748570, 18]-NRT-code), using
- digital (14, 22, 208)-net over F8, using
- (u, u+v)-construction [i] based on
(136, 136+17, 2097664)-Net in Base 8 — Constructive
(136, 153, 2097664)-net in base 8, using
- 81 times duplication [i] based on (135, 152, 2097664)-net in base 8, using
- base change [i] based on digital (97, 114, 2097664)-net over F16, using
- (u, u+v)-construction [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
- digital (81, 98, 2097150)-net over F16, using
- net defined by OOA [i] based on linear OOA(1698, 2097150, F16, 18, 17) (dual of [(2097150, 18), 37748602, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(1698, 8388601, F16, 2, 17) (dual of [(8388601, 2), 16777104, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1698, 8388602, F16, 2, 17) (dual of [(8388602, 2), 16777106, 18]-NRT-code), using
- trace code [i] based on linear OOA(25649, 4194301, F256, 2, 17) (dual of [(4194301, 2), 8388553, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25649, 8388602, F256, 17) (dual of [8388602, 8388553, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- OOA 2-folding [i] based on linear OA(25649, 8388602, F256, 17) (dual of [8388602, 8388553, 18]-code), using
- trace code [i] based on linear OOA(25649, 4194301, F256, 2, 17) (dual of [(4194301, 2), 8388553, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1698, 8388602, F16, 2, 17) (dual of [(8388602, 2), 16777106, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(1698, 8388601, F16, 2, 17) (dual of [(8388601, 2), 16777104, 18]-NRT-code), using
- net defined by OOA [i] based on linear OOA(1698, 2097150, F16, 18, 17) (dual of [(2097150, 18), 37748602, 18]-NRT-code), using
- digital (8, 16, 514)-net over F16, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (97, 114, 2097664)-net over F16, using
(136, 136+17, large)-Net over F8 — Digital
Digital (136, 153, large)-net over F8, using
- t-expansion [i] based on digital (131, 153, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8153, large, F8, 22) (dual of [large, large−153, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8153, large, F8, 22) (dual of [large, large−153, 23]-code), using
(136, 136+17, large)-Net in Base 8 — Upper bound on s
There is no (136, 153, large)-net in base 8, because
- 15 times m-reduction [i] would yield (136, 138, large)-net in base 8, but