Best Known (15, 15+8, s)-Nets in Base 8
(15, 15+8, 208)-Net over F8 — Constructive and digital
Digital (15, 23, 208)-net over F8, using
- 1 times m-reduction [i] based on digital (15, 24, 208)-net over F8, using
- trace code for nets [i] based on digital (3, 12, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- trace code for nets [i] based on digital (3, 12, 104)-net over F64, using
(15, 15+8, 514)-Net in Base 8 — Constructive
(15, 23, 514)-net in base 8, using
- 1 times m-reduction [i] based on (15, 24, 514)-net in base 8, using
- base change [i] based on digital (9, 18, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 9, 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, 9, 257)-net over F256, using
- base change [i] based on digital (9, 18, 514)-net over F16, using
(15, 15+8, 539)-Net over F8 — Digital
Digital (15, 23, 539)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(823, 539, F8, 8) (dual of [539, 516, 9]-code), using
- 21 step Varšamov–Edel lengthening with (ri) = (1, 20 times 0) [i] based on linear OA(822, 517, F8, 8) (dual of [517, 495, 9]-code), using
- construction XX applied to C1 = C([510,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([510,6]) [i] based on
- linear OA(819, 511, F8, 7) (dual of [511, 492, 8]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,5}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(819, 511, F8, 7) (dual of [511, 492, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(822, 511, F8, 8) (dual of [511, 489, 9]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,6}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(816, 511, F8, 6) (dual of [511, 495, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [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([510,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([510,6]) [i] based on
- 21 step Varšamov–Edel lengthening with (ri) = (1, 20 times 0) [i] based on linear OA(822, 517, F8, 8) (dual of [517, 495, 9]-code), using
(15, 15+8, 49283)-Net in Base 8 — Upper bound on s
There is no (15, 23, 49284)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 590 306717 844997 979008 > 823 [i]