Best Known (154, 154+17, s)-Nets in Base 8
(154, 154+17, 2105470)-Net over F8 — Constructive and digital
Digital (154, 171, 2105470)-net over F8, using
- 81 times duplication [i] based on digital (153, 170, 2105470)-net over F8, using
- trace code for nets [i] based on digital (68, 85, 1052735)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (12, 20, 4160)-net over F64, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 0, 65)-net over F64, using
- s-reduction based on digital (0, 0, s)-net over F64 with arbitrarily large s, using
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 0, 65)-net over F64 (see above)
- digital (0, 1, 65)-net over F64, using
- s-reduction based on digital (0, 1, s)-net over F64 with arbitrarily large s, using
- digital (0, 1, 65)-net over F64 (see above)
- digital (0, 1, 65)-net over F64 (see above)
- digital (0, 1, 65)-net over F64 (see above)
- digital (0, 2, 65)-net over F64, using
- digital (0, 2, 65)-net over F64 (see above)
- 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
- digital (0, 8, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64 (see above)
- digital (0, 0, 65)-net over F64, using
- generalized (u, u+v)-construction [i] based on
- digital (48, 65, 1048575)-net over F64, using
- net defined by OOA [i] based on linear OOA(6465, 1048575, F64, 17, 17) (dual of [(1048575, 17), 17825710, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(6465, 8388601, F64, 17) (dual of [8388601, 8388536, 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 8-folding and stacking with additional row [i] based on linear OA(6465, 8388601, F64, 17) (dual of [8388601, 8388536, 18]-code), using
- net defined by OOA [i] based on linear OOA(6465, 1048575, F64, 17, 17) (dual of [(1048575, 17), 17825710, 18]-NRT-code), using
- digital (12, 20, 4160)-net over F64, using
- (u, u+v)-construction [i] based on
- trace code for nets [i] based on digital (68, 85, 1052735)-net over F64, using
(154, 154+17, 2129919)-Net in Base 8 — Constructive
(154, 171, 2129919)-net in base 8, using
- 81 times duplication [i] based on (153, 170, 2129919)-net in base 8, using
- (u, u+v)-construction [i] based on
- (32, 40, 32769)-net in base 8, using
- base change [i] based on digital (22, 30, 32769)-net over F16, using
- net defined by OOA [i] based on linear OOA(1630, 32769, F16, 8, 8) (dual of [(32769, 8), 262122, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(1630, 131076, F16, 8) (dual of [131076, 131046, 9]-code), using
- trace code [i] based on linear OA(25615, 65538, F256, 8) (dual of [65538, 65523, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(25615, 65536, F256, 8) (dual of [65536, 65521, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(25613, 65536, F256, 7) (dual of [65536, 65523, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- trace code [i] based on linear OA(25615, 65538, F256, 8) (dual of [65538, 65523, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(1630, 131076, F16, 8) (dual of [131076, 131046, 9]-code), using
- net defined by OOA [i] based on linear OOA(1630, 32769, F16, 8, 8) (dual of [(32769, 8), 262122, 9]-NRT-code), using
- base change [i] based on digital (22, 30, 32769)-net over F16, 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
- (32, 40, 32769)-net in base 8, using
- (u, u+v)-construction [i] based on
(154, 154+17, large)-Net over F8 — Digital
Digital (154, 171, large)-net over F8, using
- 83 times duplication [i] based on digital (151, 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
(154, 154+17, large)-Net in Base 8 — Upper bound on s
There is no (154, 171, large)-net in base 8, because
- 15 times m-reduction [i] would yield (154, 156, large)-net in base 8, but