Best Known (15, 15+10, s)-Nets in Base 27
(15, 15+10, 224)-Net over F27 — Constructive and digital
Digital (15, 25, 224)-net over F27, using
- 1 times m-reduction [i] based on digital (15, 26, 224)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 1, 28)-net over F27, using
- s-reduction based on digital (0, 1, s)-net over F27 with arbitrarily large s, using
- digital (0, 1, 28)-net over F27 (see above)
- digital (0, 1, 28)-net over F27 (see above)
- digital (0, 2, 28)-net over F27, using
- digital (0, 2, 28)-net over F27 (see above)
- digital (0, 3, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 0 and N(F) ≥ 28, using
- the rational function field F27(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- digital (0, 5, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 11, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 1, 28)-net over F27, using
- generalized (u, u+v)-construction [i] based on
(15, 15+10, 246)-Net in Base 27 — Constructive
(15, 25, 246)-net in base 27, using
- 271 times duplication [i] based on (14, 24, 246)-net in base 27, using
- base change [i] based on digital (8, 18, 246)-net over F81, using
- 1 times m-reduction [i] based on digital (8, 19, 246)-net over F81, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 3, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- digital (0, 5, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81 (see above)
- digital (0, 11, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81 (see above)
- digital (0, 3, 82)-net over F81, using
- generalized (u, u+v)-construction [i] based on
- 1 times m-reduction [i] based on digital (8, 19, 246)-net over F81, using
- base change [i] based on digital (8, 18, 246)-net over F81, using
(15, 15+10, 1516)-Net over F27 — Digital
Digital (15, 25, 1516)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2725, 1516, F27, 10) (dual of [1516, 1491, 11]-code), using
- 778 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 16 times 0, 1, 77 times 0, 1, 236 times 0, 1, 442 times 0) [i] based on linear OA(2719, 732, F27, 10) (dual of [732, 713, 11]-code), using
- construction XX applied to C1 = C([727,7]), C2 = C([0,8]), C3 = C1 + C2 = C([0,7]), and C∩ = C1 ∩ C2 = C([727,8]) [i] based on
- linear OA(2717, 728, F27, 9) (dual of [728, 711, 10]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,7}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(2717, 728, F27, 9) (dual of [728, 711, 10]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(2719, 728, F27, 10) (dual of [728, 709, 11]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,8}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(2715, 728, F27, 8) (dual of [728, 713, 9]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([727,7]), C2 = C([0,8]), C3 = C1 + C2 = C([0,7]), and C∩ = C1 ∩ C2 = C([727,8]) [i] based on
- 778 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 16 times 0, 1, 77 times 0, 1, 236 times 0, 1, 442 times 0) [i] based on linear OA(2719, 732, F27, 10) (dual of [732, 713, 11]-code), using
(15, 15+10, 1437742)-Net in Base 27 — Upper bound on s
There is no (15, 25, 1437743)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 608268 202279 267198 536867 741616 658655 > 2725 [i]