Best Known (76−28, 76, s)-Nets in Base 8
(76−28, 76, 354)-Net over F8 — Constructive and digital
Digital (48, 76, 354)-net over F8, using
- 6 times m-reduction [i] based on digital (48, 82, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 41, 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, 41, 177)-net over F64, using
(76−28, 76, 514)-Net in Base 8 — Constructive
(48, 76, 514)-net in base 8, using
- base change [i] based on digital (29, 57, 514)-net over F16, using
- 1 times m-reduction [i] based on digital (29, 58, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 29, 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, 29, 257)-net over F256, using
- 1 times m-reduction [i] based on digital (29, 58, 514)-net over F16, using
(76−28, 76, 560)-Net over F8 — Digital
Digital (48, 76, 560)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(876, 560, F8, 28) (dual of [560, 484, 29]-code), using
- 37 step Varšamov–Edel lengthening with (ri) = (1, 8 times 0, 1, 27 times 0) [i] based on linear OA(874, 521, F8, 28) (dual of [521, 447, 29]-code), using
- construction XX applied to C1 = C([509,24]), C2 = C([0,25]), C3 = C1 + C2 = C([0,24]), and C∩ = C1 ∩ C2 = C([509,25]) [i] based on
- 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 = {−2,−1,…,24}, and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(867, 511, F8, 26) (dual of [511, 444, 27]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,25], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(873, 511, F8, 28) (dual of [511, 438, 29]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−2,−1,…,25}, and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(864, 511, F8, 25) (dual of [511, 447, 26]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,24], and designed minimum distance d ≥ |I|+1 = 26 [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(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([509,24]), C2 = C([0,25]), C3 = C1 + C2 = C([0,24]), and C∩ = C1 ∩ C2 = C([509,25]) [i] based on
- 37 step Varšamov–Edel lengthening with (ri) = (1, 8 times 0, 1, 27 times 0) [i] based on linear OA(874, 521, F8, 28) (dual of [521, 447, 29]-code), using
(76−28, 76, 68991)-Net in Base 8 — Upper bound on s
There is no (48, 76, 68992)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 431 415264 891803 028761 833794 698591 309548 629505 725698 997198 675311 310929 > 876 [i]