Best Known (40, 40+23, s)-Nets in Base 8
(40, 40+23, 354)-Net over F8 — Constructive and digital
Digital (40, 63, 354)-net over F8, using
- 3 times m-reduction [i] based on digital (40, 66, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 33, 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, 33, 177)-net over F64, using
(40, 40+23, 514)-Net in Base 8 — Constructive
(40, 63, 514)-net in base 8, using
- 1 times m-reduction [i] based on (40, 64, 514)-net in base 8, using
- base change [i] based on digital (24, 48, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 24, 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, 24, 257)-net over F256, using
- base change [i] based on digital (24, 48, 514)-net over F16, using
(40, 40+23, 531)-Net over F8 — Digital
Digital (40, 63, 531)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(863, 531, F8, 23) (dual of [531, 468, 24]-code), using
- 9 step Varšamov–Edel lengthening with (ri) = (1, 8 times 0) [i] based on linear OA(862, 521, F8, 23) (dual of [521, 459, 24]-code), using
- construction XX applied to C1 = C([509,19]), C2 = C([0,20]), C3 = C1 + C2 = C([0,19]), and C∩ = C1 ∩ C2 = C([509,20]) [i] based on
- 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(855, 511, F8, 21) (dual of [511, 456, 22]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [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 = {−2,−1,…,20}, and designed minimum distance d ≥ |I|+1 = 24 [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(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([509,19]), C2 = C([0,20]), C3 = C1 + C2 = C([0,19]), and C∩ = C1 ∩ C2 = C([509,20]) [i] based on
- 9 step Varšamov–Edel lengthening with (ri) = (1, 8 times 0) [i] based on linear OA(862, 521, F8, 23) (dual of [521, 459, 24]-code), using
(40, 40+23, 86303)-Net in Base 8 — Upper bound on s
There is no (40, 63, 86304)-net in base 8, because
- 1 times m-reduction [i] would yield (40, 62, 86304)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 98 090426 803482 733770 794453 078211 551558 954501 897243 359461 > 862 [i]