Best Known (72−27, 72, s)-Nets in Base 8
(72−27, 72, 354)-Net over F8 — Constructive and digital
Digital (45, 72, 354)-net over F8, using
- 4 times m-reduction [i] based on digital (45, 76, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 38, 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, 38, 177)-net over F64, using
(72−27, 72, 514)-Net in Base 8 — Constructive
(45, 72, 514)-net in base 8, using
- base change [i] based on digital (27, 54, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 27, 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, 27, 257)-net over F256, using
(72−27, 72, 520)-Net over F8 — Digital
Digital (45, 72, 520)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(872, 520, F8, 27) (dual of [520, 448, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(872, 522, F8, 27) (dual of [522, 450, 28]-code), using
- construction XX applied to C1 = C([48,73]), C2 = C([51,74]), C3 = C1 + C2 = C([51,73]), and C∩ = C1 ∩ C2 = C([48,74]) [i] based on
- 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 = {48,49,…,73}, and designed minimum distance d ≥ |I|+1 = 27 [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(870, 511, F8, 27) (dual of [511, 441, 28]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {48,49,…,74}, and designed minimum distance d ≥ |I|+1 = 28 [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(82, 8, F8, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,8)), using
- Reed–Solomon code RS(6,8) [i]
- 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([48,73]), C2 = C([51,74]), C3 = C1 + C2 = C([51,73]), and C∩ = C1 ∩ C2 = C([48,74]) [i] based on
- discarding factors / shortening the dual code based on linear OA(872, 522, F8, 27) (dual of [522, 450, 28]-code), using
(72−27, 72, 69265)-Net in Base 8 — Upper bound on s
There is no (45, 72, 69266)-net in base 8, because
- 1 times m-reduction [i] would yield (45, 71, 69266)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 13164 245905 987409 885900 192062 103096 717337 167027 550586 869171 820884 > 871 [i]