Best Known (53−26, 53, s)-Nets in Base 27
(53−26, 53, 164)-Net over F27 — Constructive and digital
Digital (27, 53, 164)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 20, 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 (7, 33, 82)-net over F27, using
- net from sequence [i] based on digital (7, 81)-sequence over F27 (see above)
- digital (7, 20, 82)-net over F27, using
(53−26, 53, 224)-Net in Base 27 — Constructive
(27, 53, 224)-net in base 27, using
- 3 times m-reduction [i] based on (27, 56, 224)-net in base 27, using
- base change [i] based on digital (13, 42, 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, 42, 224)-net over F81, using
(53−26, 53, 465)-Net over F27 — Digital
Digital (27, 53, 465)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2753, 465, F27, 26) (dual of [465, 412, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(2753, 737, F27, 26) (dual of [737, 684, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(22) [i] based on
- 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(2745, 729, F27, 23) (dual of [729, 684, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(272, 8, F27, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,27)), using
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- Reed–Solomon code RS(25,27) [i]
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- construction X applied to Ce(25) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2753, 737, F27, 26) (dual of [737, 684, 27]-code), using
(53−26, 53, 149270)-Net in Base 27 — Upper bound on s
There is no (27, 53, 149271)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 7282 823688 096223 645211 112513 162601 243853 663220 418102 277527 619121 487029 930463 > 2753 [i]