Best Known (23, 45, s)-Nets in Base 27
(23, 45, 152)-Net over F27 — Constructive and digital
Digital (23, 45, 152)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (6, 17, 76)-net over F27, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 6 and N(F) ≥ 76, using
- net from sequence [i] based on digital (6, 75)-sequence over F27, using
- digital (6, 28, 76)-net over F27, using
- net from sequence [i] based on digital (6, 75)-sequence over F27 (see above)
- digital (6, 17, 76)-net over F27, using
(23, 45, 172)-Net in Base 27 — Constructive
(23, 45, 172)-net in base 27, using
- 19 times m-reduction [i] based on (23, 64, 172)-net in base 27, using
- base change [i] based on digital (7, 48, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- base change [i] based on digital (7, 48, 172)-net over F81, using
(23, 45, 441)-Net over F27 — Digital
Digital (23, 45, 441)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2745, 441, F27, 22) (dual of [441, 396, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(2745, 737, F27, 22) (dual of [737, 692, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- linear OA(2743, 729, F27, 22) (dual of [729, 686, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(2737, 729, F27, 19) (dual of [729, 692, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(21) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(2745, 737, F27, 22) (dual of [737, 692, 23]-code), using
(23, 45, 135394)-Net in Base 27 — Upper bound on s
There is no (23, 45, 135395)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 25785 249781 033669 647300 529890 953372 170906 776464 832839 851699 273779 > 2745 [i]