Best Known (86−21, 86, s)-Nets in Base 8
(86−21, 86, 820)-Net over F8 — Constructive and digital
Digital (65, 86, 820)-net over F8, using
- 82 times duplication [i] based on digital (63, 84, 820)-net over F8, using
- net defined by OOA [i] based on linear OOA(884, 820, F8, 21, 21) (dual of [(820, 21), 17136, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(884, 8201, F8, 21) (dual of [8201, 8117, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(884, 8204, F8, 21) (dual of [8204, 8120, 22]-code), using
- trace code [i] based on linear OA(6442, 4102, F64, 21) (dual of [4102, 4060, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(6441, 4097, F64, 21) (dual of [4097, 4056, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(6437, 4097, F64, 19) (dual of [4097, 4060, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (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([0,10]) ⊂ C([0,9]) [i] based on
- trace code [i] based on linear OA(6442, 4102, F64, 21) (dual of [4102, 4060, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(884, 8204, F8, 21) (dual of [8204, 8120, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(884, 8201, F8, 21) (dual of [8201, 8117, 22]-code), using
- net defined by OOA [i] based on linear OOA(884, 820, F8, 21, 21) (dual of [(820, 21), 17136, 22]-NRT-code), using
(86−21, 86, 1028)-Net in Base 8 — Constructive
(65, 86, 1028)-net in base 8, using
- 82 times duplication [i] based on (63, 84, 1028)-net in base 8, using
- base change [i] based on digital (42, 63, 1028)-net over F16, using
- 161 times duplication [i] based on digital (41, 62, 1028)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (10, 20, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 10, 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, 10, 257)-net over F256, using
- digital (21, 42, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 21, 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, 21, 257)-net over F256, using
- digital (10, 20, 514)-net over F16, using
- (u, u+v)-construction [i] based on
- 161 times duplication [i] based on digital (41, 62, 1028)-net over F16, using
- base change [i] based on digital (42, 63, 1028)-net over F16, using
(86−21, 86, 9078)-Net over F8 — Digital
Digital (65, 86, 9078)-net over F8, using
(86−21, 86, large)-Net in Base 8 — Upper bound on s
There is no (65, 86, large)-net in base 8, because
- 19 times m-reduction [i] would yield (65, 67, large)-net in base 8, but