Best Known (45, 45+25, s)-Nets in Base 27
(45, 45+25, 280)-Net over F27 — Constructive and digital
Digital (45, 70, 280)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 2, 28)-net over F27, using
- digital (0, 2, 28)-net over F27 (see above)
- digital (0, 3, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 0 and N(F) ≥ 28, using
- the rational function field F27(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- digital (0, 3, 28)-net over F27 (see above)
- digital (0, 4, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 5, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 6, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 8, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 12, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 25, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
(45, 45+25, 547)-Net in Base 27 — Constructive
(45, 70, 547)-net in base 27, using
- 272 times duplication [i] based on (43, 68, 547)-net in base 27, using
- base change [i] based on digital (26, 51, 547)-net over F81, using
- 811 times duplication [i] based on digital (25, 50, 547)-net over F81, using
- net defined by OOA [i] based on linear OOA(8150, 547, F81, 25, 25) (dual of [(547, 25), 13625, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8150, 6565, F81, 25) (dual of [6565, 6515, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8150, 6567, F81, 25) (dual of [6567, 6517, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- linear OA(8149, 6562, F81, 25) (dual of [6562, 6513, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(8145, 6562, F81, 23) (dual of [6562, 6517, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8150, 6567, F81, 25) (dual of [6567, 6517, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8150, 6565, F81, 25) (dual of [6565, 6515, 26]-code), using
- net defined by OOA [i] based on linear OOA(8150, 547, F81, 25, 25) (dual of [(547, 25), 13625, 26]-NRT-code), using
- 811 times duplication [i] based on digital (25, 50, 547)-net over F81, using
- base change [i] based on digital (26, 51, 547)-net over F81, using
(45, 45+25, 5651)-Net over F27 — Digital
Digital (45, 70, 5651)-net over F27, using
(45, 45+25, large)-Net in Base 27 — Upper bound on s
There is no (45, 70, large)-net in base 27, because
- 23 times m-reduction [i] would yield (45, 47, large)-net in base 27, but