Best Known (42, 68, s)-Nets in Base 8
(42, 68, 354)-Net over F8 — Constructive and digital
Digital (42, 68, 354)-net over F8, using
- 2 times m-reduction [i] based on digital (42, 70, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 35, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- trace code for nets [i] based on digital (7, 35, 177)-net over F64, using
(42, 68, 384)-Net in Base 8 — Constructive
(42, 68, 384)-net in base 8, using
- trace code for nets [i] based on (8, 34, 192)-net in base 64, using
- 1 times m-reduction [i] based on (8, 35, 192)-net in base 64, using
- base change [i] based on digital (3, 30, 192)-net over F128, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 3 and N(F) ≥ 192, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- base change [i] based on digital (3, 30, 192)-net over F128, using
- 1 times m-reduction [i] based on (8, 35, 192)-net in base 64, using
(42, 68, 452)-Net over F8 — Digital
Digital (42, 68, 452)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(868, 452, F8, 26) (dual of [452, 384, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(868, 518, F8, 26) (dual of [518, 450, 27]-code), using
- construction XX applied to C1 = C([49,73]), C2 = C([51,74]), C3 = C1 + C2 = C([51,73]), and C∩ = C1 ∩ C2 = C([49,74]) [i] based on
- linear OA(864, 511, F8, 25) (dual of [511, 447, 26]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {49,50,…,73}, and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(864, 511, F8, 24) (dual of [511, 447, 25]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {51,52,…,74}, and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(867, 511, F8, 26) (dual of [511, 444, 27]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {49,50,…,74}, and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(861, 511, F8, 23) (dual of [511, 450, 24]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {51,52,…,73}, and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(81, 4, F8, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, 8, F8, 1) (dual of [8, 7, 2]-code), using
- Reed–Solomon code RS(7,8) [i]
- discarding factors / shortening the dual code based on linear OA(81, 8, F8, 1) (dual of [8, 7, 2]-code), using
- linear OA(80, 3, F8, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction XX applied to C1 = C([49,73]), C2 = C([51,74]), C3 = C1 + C2 = C([51,73]), and C∩ = C1 ∩ C2 = C([49,74]) [i] based on
- discarding factors / shortening the dual code based on linear OA(868, 518, F8, 26) (dual of [518, 450, 27]-code), using
(42, 68, 42863)-Net in Base 8 — Upper bound on s
There is no (42, 68, 42864)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 25 717811 138376 451100 292922 756856 250877 673652 324143 906190 608955 > 868 [i]