Best Known (25−11, 25, s)-Nets in Base 27
(25−11, 25, 196)-Net over F27 — Constructive and digital
Digital (14, 25, 196)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 1, 28)-net over F27, using
- s-reduction based on digital (0, 1, s)-net over F27 with arbitrarily large s, using
- digital (0, 1, 28)-net over F27 (see above)
- 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, 5, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 11, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 1, 28)-net over F27, using
(25−11, 25, 232)-Net in Base 27 — Constructive
(14, 25, 232)-net in base 27, using
- (u, u+v)-construction [i] based on
- digital (2, 7, 351)-net over F27, using
- net defined by OOA [i] based on linear OOA(277, 351, F27, 5, 5) (dual of [(351, 5), 1748, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(277, 703, F27, 5) (dual of [703, 696, 6]-code), using
- net defined by OOA [i] based on linear OOA(277, 351, F27, 5, 5) (dual of [(351, 5), 1748, 6]-NRT-code), using
- (7, 18, 116)-net in base 27, using
- 2 times m-reduction [i] based on (7, 20, 116)-net in base 27, using
- base change [i] based on digital (2, 15, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- base change [i] based on digital (2, 15, 116)-net over F81, using
- 2 times m-reduction [i] based on (7, 20, 116)-net in base 27, using
- digital (2, 7, 351)-net over F27, using
(25−11, 25, 793)-Net over F27 — Digital
Digital (14, 25, 793)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2725, 793, F27, 11) (dual of [793, 768, 12]-code), using
- 55 step Varšamov–Edel lengthening with (ri) = (2, 8 times 0, 1, 45 times 0) [i] based on linear OA(2722, 735, F27, 11) (dual of [735, 713, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(2721, 730, F27, 11) (dual of [730, 709, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(2717, 730, F27, 9) (dual of [730, 713, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(271, 5, F27, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, s, F27, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- 55 step Varšamov–Edel lengthening with (ri) = (2, 8 times 0, 1, 45 times 0) [i] based on linear OA(2722, 735, F27, 11) (dual of [735, 713, 12]-code), using
(25−11, 25, 743716)-Net in Base 27 — Upper bound on s
There is no (14, 25, 743717)-net in base 27, because
- 1 times m-reduction [i] would yield (14, 24, 743717)-net in base 27, but
- the generalized Rao bound for nets shows that 27m ≥ 22528 404672 236354 582302 977191 046195 > 2724 [i]