Best Known (31, 61, s)-Nets in Base 27
(31, 61, 170)-Net over F27 — Constructive and digital
Digital (31, 61, 170)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 22, 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 (9, 39, 88)-net over F27, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 9 and N(F) ≥ 88, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- digital (7, 22, 82)-net over F27, using
(31, 61, 224)-Net in Base 27 — Constructive
(31, 61, 224)-net in base 27, using
- 11 times m-reduction [i] based on (31, 72, 224)-net in base 27, using
- base change [i] based on digital (13, 54, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- base change [i] based on digital (13, 54, 224)-net over F81, using
(31, 61, 495)-Net over F27 — Digital
Digital (31, 61, 495)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2761, 495, F27, 30) (dual of [495, 434, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(2761, 743, F27, 30) (dual of [743, 682, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(23) [i] based on
- linear OA(2756, 729, F27, 30) (dual of [729, 673, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(2747, 729, F27, 24) (dual of [729, 682, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(275, 14, F27, 5) (dual of [14, 9, 6]-code or 14-arc in PG(4,27)), using
- discarding factors / shortening the dual code based on linear OA(275, 27, F27, 5) (dual of [27, 22, 6]-code or 27-arc in PG(4,27)), using
- Reed–Solomon code RS(22,27) [i]
- discarding factors / shortening the dual code based on linear OA(275, 27, F27, 5) (dual of [27, 22, 6]-code or 27-arc in PG(4,27)), using
- construction X applied to Ce(29) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(2761, 743, F27, 30) (dual of [743, 682, 31]-code), using
(31, 61, 163550)-Net in Base 27 — Upper bound on s
There is no (31, 61, 163551)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 2056 839734 537754 272104 260433 363702 144906 706945 542349 408735 443692 972907 855752 020738 379003 > 2761 [i]