Best Known (39−16, 39, s)-Nets in Base 27
(39−16, 39, 192)-Net over F27 — Constructive and digital
Digital (23, 39, 192)-net over F27, using
- 1 times m-reduction [i] based on digital (23, 40, 192)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (2, 7, 64)-net over F27, using
- s-reduction based on digital (2, 7, 351)-net over F27, using
- net defined by OOA [i] based on linear OOA(277, 351, F27, 5, 5) (dual of [(351, 5), 1748, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(277, 703, F27, 5) (dual of [703, 696, 6]-code), using
- net defined by OOA [i] based on linear OOA(277, 351, F27, 5, 5) (dual of [(351, 5), 1748, 6]-NRT-code), using
- s-reduction based on digital (2, 7, 351)-net over F27, using
- digital (4, 12, 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 (4, 21, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27 (see above)
- digital (2, 7, 64)-net over F27, using
- generalized (u, u+v)-construction [i] based on
(39−16, 39, 250)-Net in Base 27 — Constructive
(23, 39, 250)-net in base 27, using
- 1 times m-reduction [i] based on (23, 40, 250)-net in base 27, using
- base change [i] based on digital (13, 30, 250)-net over F81, using
- (u, u+v)-construction [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
- digital (4, 21, 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, 9, 100)-net over F81, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (13, 30, 250)-net over F81, using
(39−16, 39, 1310)-Net over F27 — Digital
Digital (23, 39, 1310)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2739, 1310, F27, 16) (dual of [1310, 1271, 17]-code), using
- 570 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 12 times 0, 1, 36 times 0, 1, 92 times 0, 1, 173 times 0, 1, 248 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
- 570 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 0, 1, 12 times 0, 1, 36 times 0, 1, 92 times 0, 1, 173 times 0, 1, 248 times 0) [i] based on linear OA(2731, 732, F27, 16) (dual of [732, 701, 17]-code), using
(39−16, 39, 1375985)-Net in Base 27 — Upper bound on s
There is no (23, 39, 1375986)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 66 556120 011234 475145 959853 942736 308093 546135 825127 380817 > 2739 [i]