Best Known (26−14, 26, s)-Nets in Base 27
(26−14, 26, 102)-Net over F27 — Constructive and digital
Digital (12, 26, 102)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (1, 8, 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
- digital (4, 18, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 4 and N(F) ≥ 64, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- digital (1, 8, 38)-net over F27, using
(26−14, 26, 160)-Net in Base 27 — Constructive
(12, 26, 160)-net in base 27, using
- 2 times m-reduction [i] based on (12, 28, 160)-net in base 27, using
- base change [i] based on digital (5, 21, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- base change [i] based on digital (5, 21, 160)-net over F81, using
(26−14, 26, 186)-Net over F27 — Digital
Digital (12, 26, 186)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2726, 186, F27, 14) (dual of [186, 160, 15]-code), using
- construction XX applied to C1 = C([181,11]), C2 = C([0,12]), C3 = C1 + C2 = C([0,11]), and C∩ = C1 ∩ C2 = C([181,12]) [i] based on
- linear OA(2724, 182, F27, 13) (dual of [182, 158, 14]-code), using the BCH-code C(I) with length 182 | 272−1, defining interval I = {−1,0,…,11}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(2724, 182, F27, 13) (dual of [182, 158, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 182 | 272−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(2726, 182, F27, 14) (dual of [182, 156, 15]-code), using the BCH-code C(I) with length 182 | 272−1, defining interval I = {−1,0,…,12}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2722, 182, F27, 12) (dual of [182, 160, 13]-code), using the expurgated narrow-sense BCH-code C(I) with length 182 | 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([181,11]), C2 = C([0,12]), C3 = C1 + C2 = C([0,11]), and C∩ = C1 ∩ C2 = C([181,12]) [i] based on
(26−14, 26, 26939)-Net in Base 27 — Upper bound on s
There is no (12, 26, 26940)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 16 426117 427772 849985 909452 547533 209649 > 2726 [i]