Best Known (80−30, 80, s)-Nets in Base 8
(80−30, 80, 354)-Net over F8 — Constructive and digital
Digital (50, 80, 354)-net over F8, using
- 6 times m-reduction [i] based on digital (50, 86, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 43, 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, 43, 177)-net over F64, using
(80−30, 80, 514)-Net in Base 8 — Constructive
(50, 80, 514)-net in base 8, using
- base change [i] based on digital (30, 60, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 30, 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, 30, 257)-net over F256, using
(80−30, 80, 536)-Net over F8 — Digital
Digital (50, 80, 536)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(880, 536, F8, 30) (dual of [536, 456, 31]-code), using
- 18 step Varšamov–Edel lengthening with (ri) = (1, 17 times 0) [i] based on linear OA(879, 517, F8, 30) (dual of [517, 438, 31]-code), using
- construction XX applied to C1 = C([510,27]), C2 = C([0,28]), C3 = C1 + C2 = C([0,27]), and C∩ = C1 ∩ C2 = C([510,28]) [i] based on
- linear OA(876, 511, F8, 29) (dual of [511, 435, 30]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,27}, and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(876, 511, F8, 29) (dual of [511, 435, 30]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,28], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(879, 511, F8, 30) (dual of [511, 432, 31]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,28}, and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(873, 511, F8, 28) (dual of [511, 438, 29]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,27], and designed minimum distance d ≥ |I|+1 = 29 [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
- linear OA(80, 3, F8, 0) (dual of [3, 3, 1]-code) (see above)
- construction XX applied to C1 = C([510,27]), C2 = C([0,28]), C3 = C1 + C2 = C([0,27]), and C∩ = C1 ∩ C2 = C([510,28]) [i] based on
- 18 step Varšamov–Edel lengthening with (ri) = (1, 17 times 0) [i] based on linear OA(879, 517, F8, 30) (dual of [517, 438, 31]-code), using
(80−30, 80, 60128)-Net in Base 8 — Upper bound on s
There is no (50, 80, 60129)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 1 766937 253809 970472 307654 667907 433278 743247 012016 930288 636252 690145 580512 > 880 [i]