Best Known (27−10, 27, s)-Nets in Base 5
(27−10, 27, 104)-Net over F5 — Constructive and digital
Digital (17, 27, 104)-net over F5, using
- 1 times m-reduction [i] based on digital (17, 28, 104)-net over F5, using
- trace code for nets [i] based on digital (3, 14, 52)-net over F25, using
- net from sequence [i] based on digital (3, 51)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 3 and N(F) ≥ 52, using
- net from sequence [i] based on digital (3, 51)-sequence over F25, using
- trace code for nets [i] based on digital (3, 14, 52)-net over F25, using
(27−10, 27, 143)-Net over F5 — Digital
Digital (17, 27, 143)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(527, 143, F5, 10) (dual of [143, 116, 11]-code), using
- 8 step Varšamov–Edel lengthening with (ri) = (1, 7 times 0) [i] based on linear OA(526, 134, F5, 10) (dual of [134, 108, 11]-code), using
- construction XX applied to C1 = C([122,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([122,7]) [i] based on
- linear OA(522, 124, F5, 9) (dual of [124, 102, 10]-code), using the primitive BCH-code C(I) with length 124 = 53−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(519, 124, F5, 8) (dual of [124, 105, 9]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 124 = 53−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(525, 124, F5, 10) (dual of [124, 99, 11]-code), using the primitive BCH-code C(I) with length 124 = 53−1, defining interval I = {−2,−1,…,7}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(516, 124, F5, 7) (dual of [124, 108, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 124 = 53−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(51, 7, F5, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(50, 3, F5, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction XX applied to C1 = C([122,6]), C2 = C([0,7]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([122,7]) [i] based on
- 8 step Varšamov–Edel lengthening with (ri) = (1, 7 times 0) [i] based on linear OA(526, 134, F5, 10) (dual of [134, 108, 11]-code), using
(27−10, 27, 3871)-Net in Base 5 — Upper bound on s
There is no (17, 27, 3872)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 7 457903 146230 218881 > 527 [i]