Best Known (31−12, 31, s)-Nets in Base 5
(31−12, 31, 104)-Net over F5 — Constructive and digital
Digital (19, 31, 104)-net over F5, using
- 1 times m-reduction [i] based on digital (19, 32, 104)-net over F5, using
- trace code for nets [i] based on digital (3, 16, 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, 16, 52)-net over F25, using
(31−12, 31, 136)-Net over F5 — Digital
Digital (19, 31, 136)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(531, 136, F5, 12) (dual of [136, 105, 13]-code), using
- construction XX applied to C1 = C([21,31]), C2 = C([24,32]), C3 = C1 + C2 = C([24,31]), 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(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 = {24,25,…,32}, and designed minimum distance d ≥ |I|+1 = 10 [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(519, 124, F5, 8) (dual of [124, 105, 9]-code), using the primitive BCH-code C(I) with length 124 = 53−1, defining interval I = {24,25,…,31}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(53, 9, F5, 2) (dual of [9, 6, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 24 = 52−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-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([24,32]), C3 = C1 + C2 = C([24,31]), and C∩ = C1 ∩ C2 = C([21,32]) [i] based on
(31−12, 31, 3054)-Net in Base 5 — Upper bound on s
There is no (19, 31, 3055)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 4661 252216 383203 609065 > 531 [i]