Best Known (36−15, 36, s)-Nets in Base 27
(36−15, 36, 178)-Net over F27 — Constructive and digital
Digital (21, 36, 178)-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, 3, 28)-net over F27 (see above)
- digital (0, 5, 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, 16, 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
(36−15, 36, 250)-Net in Base 27 — Constructive
(21, 36, 250)-net in base 27, using
- base change [i] based on digital (12, 27, 250)-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 (4, 19, 150)-net over F81, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 4 and N(F) ≥ 150, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- digital (1, 8, 100)-net over F81, using
- (u, u+v)-construction [i] based on
(36−15, 36, 1128)-Net over F27 — Digital
Digital (21, 36, 1128)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2736, 1128, F27, 15) (dual of [1128, 1092, 16]-code), using
- 389 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 16 times 0, 1, 47 times 0, 1, 114 times 0, 1, 204 times 0) [i] based on linear OA(2729, 732, F27, 15) (dual of [732, 703, 16]-code), using
- construction XX applied to C1 = C([727,12]), C2 = C([0,13]), C3 = C1 + C2 = C([0,12]), and C∩ = C1 ∩ C2 = C([727,13]) [i] based on
- 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(2727, 728, F27, 14) (dual of [728, 701, 15]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2729, 728, F27, 15) (dual of [728, 699, 16]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,13}, and designed minimum distance d ≥ |I|+1 = 16 [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(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([727,12]), C2 = C([0,13]), C3 = C1 + C2 = C([0,12]), and C∩ = C1 ∩ C2 = C([727,13]) [i] based on
- 389 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 16 times 0, 1, 47 times 0, 1, 114 times 0, 1, 204 times 0) [i] based on linear OA(2729, 732, F27, 15) (dual of [732, 703, 16]-code), using
(36−15, 36, 1865363)-Net in Base 27 — Upper bound on s
There is no (21, 36, 1865364)-net in base 27, because
- 1 times m-reduction [i] would yield (21, 35, 1865364)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 125 237204 301728 039817 872221 547854 341756 729261 595281 > 2735 [i]