Best Known (51, 68, s)-Nets in Base 8
(51, 68, 1025)-Net over F8 — Constructive and digital
Digital (51, 68, 1025)-net over F8, using
- net defined by OOA [i] based on linear OOA(868, 1025, F8, 17, 17) (dual of [(1025, 17), 17357, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(868, 8201, F8, 17) (dual of [8201, 8133, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(868, 8204, F8, 17) (dual of [8204, 8136, 18]-code), using
- trace code [i] based on linear OA(6434, 4102, F64, 17) (dual of [4102, 4068, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- linear OA(6433, 4097, F64, 17) (dual of [4097, 4064, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(6429, 4097, F64, 15) (dual of [4097, 4068, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- trace code [i] based on linear OA(6434, 4102, F64, 17) (dual of [4102, 4068, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(868, 8204, F8, 17) (dual of [8204, 8136, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(868, 8201, F8, 17) (dual of [8201, 8133, 18]-code), using
(51, 68, 1028)-Net in Base 8 — Constructive
(51, 68, 1028)-net in base 8, using
- base change [i] based on digital (34, 51, 1028)-net over F16, using
- 161 times duplication [i] based on digital (33, 50, 1028)-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 (17, 34, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 17, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- trace code for nets [i] based on digital (0, 17, 257)-net over F256, using
- digital (8, 16, 514)-net over F16, using
- (u, u+v)-construction [i] based on
- 161 times duplication [i] based on digital (33, 50, 1028)-net over F16, using
(51, 68, 8204)-Net over F8 — Digital
Digital (51, 68, 8204)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(868, 8204, F8, 17) (dual of [8204, 8136, 18]-code), using
- trace code [i] based on linear OA(6434, 4102, F64, 17) (dual of [4102, 4068, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- linear OA(6433, 4097, F64, 17) (dual of [4097, 4064, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(6429, 4097, F64, 15) (dual of [4097, 4068, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- trace code [i] based on linear OA(6434, 4102, F64, 17) (dual of [4102, 4068, 18]-code), using
(51, 68, large)-Net in Base 8 — Upper bound on s
There is no (51, 68, large)-net in base 8, because
- 15 times m-reduction [i] would yield (51, 53, large)-net in base 8, but