Best Known (25, 48, s)-Nets in Base 27
(25, 48, 164)-Net over F27 — Constructive and digital
Digital (25, 48, 164)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 18, 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, 30, 82)-net over F27, using
- net from sequence [i] based on digital (7, 81)-sequence over F27 (see above)
- digital (7, 18, 82)-net over F27, using
(25, 48, 224)-Net in Base 27 — Constructive
(25, 48, 224)-net in base 27, using
- base change [i] based on digital (13, 36, 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
(25, 48, 524)-Net over F27 — Digital
Digital (25, 48, 524)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2748, 524, F27, 23) (dual of [524, 476, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2748, 741, F27, 23) (dual of [741, 693, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,9]) [i] based on
- linear OA(2745, 730, F27, 23) (dual of [730, 685, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(2737, 730, F27, 19) (dual of [730, 693, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(273, 11, F27, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,27) or 11-cap in PG(2,27)), using
- discarding factors / shortening the dual code based on linear OA(273, 27, F27, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
- Reed–Solomon code RS(24,27) [i]
- discarding factors / shortening the dual code based on linear OA(273, 27, F27, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
- construction X applied to C([0,11]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2748, 741, F27, 23) (dual of [741, 693, 24]-code), using
(25, 48, 246523)-Net in Base 27 — Upper bound on s
There is no (25, 48, 246524)-net in base 27, because
- 1 times m-reduction [i] would yield (25, 47, 246524)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 18 797818 945308 119037 184847 731262 955519 478165 659831 101194 542311 191665 > 2747 [i]