Best Known (60−22, 60, s)-Nets in Base 8
(60−22, 60, 354)-Net over F8 — Constructive and digital
Digital (38, 60, 354)-net over F8, using
- 2 times m-reduction [i] based on digital (38, 62, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 31, 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, 31, 177)-net over F64, using
(60−22, 60, 514)-Net in Base 8 — Constructive
(38, 60, 514)-net in base 8, using
- base change [i] based on digital (23, 45, 514)-net over F16, using
- 1 times m-reduction [i] based on digital (23, 46, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 23, 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, 23, 257)-net over F256, using
- 1 times m-reduction [i] based on digital (23, 46, 514)-net over F16, using
(60−22, 60, 525)-Net over F8 — Digital
Digital (38, 60, 525)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(860, 525, F8, 22) (dual of [525, 465, 23]-code), using
- construction XX applied to C1 = C([509,17]), C2 = C([0,19]), C3 = C1 + C2 = C([0,17]), and C∩ = C1 ∩ C2 = C([509,19]) [i] based on
- linear OA(852, 511, F8, 20) (dual of [511, 459, 21]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−2,−1,…,17}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(852, 511, F8, 20) (dual of [511, 459, 21]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(858, 511, F8, 22) (dual of [511, 453, 23]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−2,−1,…,19}, and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(846, 511, F8, 18) (dual of [511, 465, 19]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(81, 7, F8, 1) (dual of [7, 6, 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(81, 7, F8, 1) (dual of [7, 6, 2]-code) (see above)
- construction XX applied to C1 = C([509,17]), C2 = C([0,19]), C3 = C1 + C2 = C([0,17]), and C∩ = C1 ∩ C2 = C([509,19]) [i] based on
(60−22, 60, 59130)-Net in Base 8 — Upper bound on s
There is no (38, 60, 59131)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 1 532572 557642 541131 623474 088370 912552 729063 970381 491928 > 860 [i]