Best Known (13, 21, s)-Nets in Base 5
(13, 21, 64)-Net over F5 — Constructive and digital
Digital (13, 21, 64)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 12)-net over F5, using
- digital (8, 16, 52)-net over F5, using
- trace code for nets [i] based on digital (0, 8, 26)-net over F25, using
- net from sequence [i] based on digital (0, 25)-sequence over F25, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26, using
- the rational function field F25(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 25)-sequence over F25, using
- trace code for nets [i] based on digital (0, 8, 26)-net over F25, using
(13, 21, 136)-Net over F5 — Digital
Digital (13, 21, 136)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(521, 136, F5, 8) (dual of [136, 115, 9]-code), using
- 4 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0) [i] based on linear OA(519, 130, F5, 8) (dual of [130, 111, 9]-code), using
- construction XX applied to C1 = C([123,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([123,6]) [i] based on
- linear OA(516, 124, F5, 7) (dual of [124, 108, 8]-code), using the primitive BCH-code C(I) with length 124 = 53−1, defining interval I = {−1,0,…,5}, and designed minimum distance d ≥ |I|+1 = 8 [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(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 = {−1,0,…,6}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(513, 124, F5, 6) (dual of [124, 111, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 124 = 53−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- 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
- linear OA(50, 3, F5, 0) (dual of [3, 3, 1]-code) (see above)
- construction XX applied to C1 = C([123,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([123,6]) [i] based on
- 4 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0) [i] based on linear OA(519, 130, F5, 8) (dual of [130, 111, 9]-code), using
(13, 21, 2583)-Net in Base 5 — Upper bound on s
There is no (13, 21, 2584)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 477 395153 332225 > 521 [i]