Best Known (37, 37+34, s)-Nets in Base 27
(37, 37+34, 188)-Net over F27 — Constructive and digital
Digital (37, 71, 188)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (10, 27, 94)-net over F27, using
- net from sequence [i] based on digital (10, 93)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 10 and N(F) ≥ 94, using
- net from sequence [i] based on digital (10, 93)-sequence over F27, using
- digital (10, 44, 94)-net over F27, using
- net from sequence [i] based on digital (10, 93)-sequence over F27 (see above)
- digital (10, 27, 94)-net over F27, using
(37, 37+34, 370)-Net in Base 27 — Constructive
(37, 71, 370)-net in base 27, using
- 13 times m-reduction [i] based on (37, 84, 370)-net in base 27, using
- base change [i] based on digital (16, 63, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- base change [i] based on digital (16, 63, 370)-net over F81, using
(37, 37+34, 651)-Net over F27 — Digital
Digital (37, 71, 651)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2771, 651, F27, 34) (dual of [651, 580, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(2771, 749, F27, 34) (dual of [749, 678, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(25) [i] based on
- linear OA(2764, 729, F27, 34) (dual of [729, 665, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(2751, 729, F27, 26) (dual of [729, 678, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(277, 20, F27, 7) (dual of [20, 13, 8]-code or 20-arc in PG(6,27)), using
- discarding factors / shortening the dual code based on linear OA(277, 27, F27, 7) (dual of [27, 20, 8]-code or 27-arc in PG(6,27)), using
- Reed–Solomon code RS(20,27) [i]
- discarding factors / shortening the dual code based on linear OA(277, 27, F27, 7) (dual of [27, 20, 8]-code or 27-arc in PG(6,27)), using
- construction X applied to Ce(33) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(2771, 749, F27, 34) (dual of [749, 678, 35]-code), using
(37, 37+34, 262425)-Net in Base 27 — Upper bound on s
There is no (37, 71, 262426)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 423485 448973 544895 020214 358674 808543 505229 911253 032124 584930 419488 076166 776730 604508 052237 864026 162437 > 2771 [i]