Best Known (75−28, 75, s)-Nets in Base 8
(75−28, 75, 354)-Net over F8 — Constructive and digital
Digital (47, 75, 354)-net over F8, using
- 5 times m-reduction [i] based on digital (47, 80, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 40, 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, 40, 177)-net over F64, using
(75−28, 75, 384)-Net in Base 8 — Constructive
(47, 75, 384)-net in base 8, using
- 1 times m-reduction [i] based on (47, 76, 384)-net in base 8, using
- trace code for nets [i] based on (9, 38, 192)-net in base 64, using
- 4 times m-reduction [i] based on (9, 42, 192)-net in base 64, using
- base change [i] based on digital (3, 36, 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, 36, 192)-net over F128, using
- 4 times m-reduction [i] based on (9, 42, 192)-net in base 64, using
- trace code for nets [i] based on (9, 38, 192)-net in base 64, using
(75−28, 75, 531)-Net over F8 — Digital
Digital (47, 75, 531)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(875, 531, F8, 28) (dual of [531, 456, 29]-code), using
- 9 step Varšamov–Edel lengthening with (ri) = (1, 8 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
- 9 step Varšamov–Edel lengthening with (ri) = (1, 8 times 0) [i] based on linear OA(874, 521, F8, 28) (dual of [521, 447, 29]-code), using
(75−28, 75, 59467)-Net in Base 8 — Upper bound on s
There is no (47, 75, 59468)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 53 926200 420553 677656 138909 088176 490849 521676 847045 905026 718680 531088 > 875 [i]