Best Known (15, 15+12, s)-Nets in Base 27
(15, 15+12, 140)-Net over F27 — Constructive and digital
Digital (15, 27, 140)-net over F27, using
- 1 times m-reduction [i] based on digital (15, 28, 140)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 2, 28)-net over F27, using
- 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, 4, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 6, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 13, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- generalized (u, u+v)-construction [i] based on
(15, 15+12, 200)-Net in Base 27 — Constructive
(15, 27, 200)-net in base 27, using
- 1 times m-reduction [i] based on (15, 28, 200)-net in base 27, using
- base change [i] based on digital (8, 21, 200)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- digital (1, 14, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81 (see above)
- digital (1, 7, 100)-net over F81, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (8, 21, 200)-net over F81, using
(15, 15+12, 771)-Net over F27 — Digital
Digital (15, 27, 771)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2727, 771, F27, 12) (dual of [771, 744, 13]-code), using
- 33 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 27 times 0) [i] based on linear OA(2724, 735, F27, 12) (dual of [735, 711, 13]-code), using
- construction XX applied to C1 = C([726,8]), C2 = C([0,9]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([726,9]) [i] based on
- linear OA(2721, 728, F27, 11) (dual of [728, 707, 12]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−2,−1,…,8}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2719, 728, F27, 10) (dual of [728, 709, 11]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(2723, 728, F27, 12) (dual of [728, 705, 13]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−2,−1,…,9}, and designed minimum distance d ≥ |I|+1 = 13 [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(271, 5, F27, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, 27, F27, 1) (dual of [27, 26, 2]-code), using
- Reed–Solomon code RS(26,27) [i]
- discarding factors / shortening the dual code based on linear OA(271, 27, F27, 1) (dual of [27, 26, 2]-code), using
- 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
- construction XX applied to C1 = C([726,8]), C2 = C([0,9]), C3 = C1 + C2 = C([0,8]), and C∩ = C1 ∩ C2 = C([726,9]) [i] based on
- 33 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 27 times 0) [i] based on linear OA(2724, 735, F27, 12) (dual of [735, 711, 13]-code), using
(15, 15+12, 317966)-Net in Base 27 — Upper bound on s
There is no (15, 27, 317967)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 443 427135 517150 732859 988030 339927 520005 > 2727 [i]