Best Known (21, 37, s)-Nets in Base 27
(21, 37, 164)-Net over F27 — Constructive and digital
Digital (21, 37, 164)-net over F27, using
- 1 times m-reduction [i] based on digital (21, 38, 164)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (6, 14, 82)-net over F27, using
- s-reduction based on digital (6, 14, 84)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 2, 28)-net over F27, using
- digital (0, 4, 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, 8, 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
- s-reduction based on digital (6, 14, 84)-net over F27, using
- digital (7, 24, 82)-net over F27, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 7 and N(F) ≥ 82, using
- net from sequence [i] based on digital (7, 81)-sequence over F27, using
- digital (6, 14, 82)-net over F27, using
- (u, u+v)-construction [i] based on
(21, 37, 216)-Net in Base 27 — Constructive
(21, 37, 216)-net in base 27, using
- 1 times m-reduction [i] based on (21, 38, 216)-net in base 27, using
- (u, u+v)-construction [i] based on
- (4, 12, 100)-net in base 27, using
- base change [i] based on digital (1, 9, 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
- base change [i] based on digital (1, 9, 100)-net over F81, using
- (9, 26, 116)-net in base 27, using
- 2 times m-reduction [i] based on (9, 28, 116)-net in base 27, using
- base change [i] based on digital (2, 21, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- base change [i] based on digital (2, 21, 116)-net over F81, using
- 2 times m-reduction [i] based on (9, 28, 116)-net in base 27, using
- (4, 12, 100)-net in base 27, using
- (u, u+v)-construction [i] based on
(21, 37, 885)-Net over F27 — Digital
Digital (21, 37, 885)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2737, 885, F27, 16) (dual of [885, 848, 17]-code), using
- 147 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 12 times 0, 1, 36 times 0, 1, 92 times 0) [i] based on linear OA(2731, 732, F27, 16) (dual of [732, 701, 17]-code), using
- construction XX applied to C1 = C([727,13]), C2 = C([0,14]), C3 = C1 + C2 = C([0,13]), and C∩ = C1 ∩ C2 = C([727,14]) [i] based on
- 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(2729, 728, F27, 15) (dual of [728, 699, 16]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(2731, 728, F27, 16) (dual of [728, 697, 17]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,14}, and designed minimum distance d ≥ |I|+1 = 17 [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(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([727,13]), C2 = C([0,14]), C3 = C1 + C2 = C([0,13]), and C∩ = C1 ∩ C2 = C([727,14]) [i] based on
- 147 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 12 times 0, 1, 36 times 0, 1, 92 times 0) [i] based on linear OA(2731, 732, F27, 16) (dual of [732, 701, 17]-code), using
(21, 37, 603630)-Net in Base 27 — Upper bound on s
There is no (21, 37, 603631)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 91297 830674 705960 478914 939765 870410 959492 033316 375089 > 2737 [i]