Best Known (17, 17+14, s)-Nets in Base 27
(17, 17+14, 150)-Net over F27 — Constructive and digital
Digital (17, 31, 150)-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, 7, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (1, 15, 38)-net over F27, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 1 and N(F) ≥ 38, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
(17, 17+14, 200)-Net in Base 27 — Constructive
(17, 31, 200)-net in base 27, using
- 1 times m-reduction [i] based on (17, 32, 200)-net in base 27, using
- base change [i] based on digital (9, 24, 200)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (1, 8, 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, 16, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81 (see above)
- digital (1, 8, 100)-net over F81, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (9, 24, 200)-net over F81, using
(17, 17+14, 762)-Net over F27 — Digital
Digital (17, 31, 762)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2731, 762, F27, 14) (dual of [762, 731, 15]-code), using
- 26 step Varšamov–Edel lengthening with (ri) = (3, 4 times 0, 1, 20 times 0) [i] based on linear OA(2727, 732, F27, 14) (dual of [732, 705, 15]-code), using
- construction XX applied to C1 = C([727,11]), C2 = C([0,12]), C3 = C1 + C2 = C([0,11]), and C∩ = C1 ∩ C2 = C([727,12]) [i] based on
- linear OA(2725, 728, F27, 13) (dual of [728, 703, 14]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,11}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(2725, 728, F27, 13) (dual of [728, 703, 14]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(2727, 728, F27, 14) (dual of [728, 701, 15]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,12}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2723, 728, F27, 12) (dual of [728, 705, 13]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [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([727,11]), C2 = C([0,12]), C3 = C1 + C2 = C([0,11]), and C∩ = C1 ∩ C2 = C([727,12]) [i] based on
- 26 step Varšamov–Edel lengthening with (ri) = (3, 4 times 0, 1, 20 times 0) [i] based on linear OA(2727, 732, F27, 14) (dual of [732, 705, 15]-code), using
(17, 17+14, 283685)-Net in Base 27 — Upper bound on s
There is no (17, 31, 283686)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 235 658581 542219 991097 909942 832779 739006 380841 > 2731 [i]