Best Known (17, 29, s)-Nets in Base 5
(17, 29, 56)-Net over F5 — Constructive and digital
Digital (17, 29, 56)-net over F5, using
- 1 times m-reduction [i] based on digital (17, 30, 56)-net over F5, using
- trace code for nets [i] based on digital (2, 15, 28)-net over F25, using
- net from sequence [i] based on digital (2, 27)-sequence over F25, using
- trace code for nets [i] based on digital (2, 15, 28)-net over F25, using
(17, 29, 97)-Net over F5 — Digital
Digital (17, 29, 97)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(529, 97, F5, 12) (dual of [97, 68, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(529, 131, F5, 12) (dual of [131, 102, 13]-code), using
- construction XX applied to C1 = C([21,31]), C2 = C([23,32]), C3 = C1 + C2 = C([23,31]), and C∩ = C1 ∩ C2 = C([21,32]) [i] based on
- linear OA(525, 124, F5, 11) (dual of [124, 99, 12]-code), using the primitive BCH-code C(I) with length 124 = 53−1, defining interval I = {21,22,…,31}, and designed minimum distance d ≥ |I|+1 = 12 [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 = {23,24,…,32}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(528, 124, F5, 12) (dual of [124, 96, 13]-code), using the primitive BCH-code C(I) with length 124 = 53−1, defining interval I = {21,22,…,32}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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 = {23,24,…,31}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(51, 4, F5, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, 5, F5, 1) (dual of [5, 4, 2]-code), using
- Reed–Solomon code RS(4,5) [i]
- discarding factors / shortening the dual code based on linear OA(51, 5, F5, 1) (dual of [5, 4, 2]-code), 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([21,31]), C2 = C([23,32]), C3 = C1 + C2 = C([23,31]), and C∩ = C1 ∩ C2 = C([21,32]) [i] based on
- discarding factors / shortening the dual code based on linear OA(529, 131, F5, 12) (dual of [131, 102, 13]-code), using
(17, 29, 1784)-Net in Base 5 — Upper bound on s
There is no (17, 29, 1785)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 186 502430 297288 485129 > 529 [i]