Best Known (45, 45+13, s)-Nets in Base 27
(45, 45+13, 88630)-Net over F27 — Constructive and digital
Digital (45, 58, 88630)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (3, 9, 56)-net over F27, using
- (u, u+v)-construction [i] based on
- 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, 6, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 3, 28)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (36, 49, 88574)-net over F27, using
- net defined by OOA [i] based on linear OOA(2749, 88574, F27, 13, 13) (dual of [(88574, 13), 1151413, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2749, 531445, F27, 13) (dual of [531445, 531396, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(2749, 531441, F27, 13) (dual of [531441, 531392, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2745, 531441, F27, 12) (dual of [531441, 531396, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(270, 4, F27, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(2749, 531445, F27, 13) (dual of [531445, 531396, 14]-code), using
- net defined by OOA [i] based on linear OOA(2749, 88574, F27, 13, 13) (dual of [(88574, 13), 1151413, 14]-NRT-code), using
- digital (3, 9, 56)-net over F27, using
(45, 45+13, 88657)-Net in Base 27 — Constructive
(45, 58, 88657)-net in base 27, using
- (u, u+v)-construction [i] based on
- (2, 8, 82)-net in base 27, using
- base change [i] based on digital (0, 6, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- base change [i] based on digital (0, 6, 82)-net over F81, using
- digital (37, 50, 88575)-net over F27, using
- net defined by OOA [i] based on linear OOA(2750, 88575, F27, 13, 13) (dual of [(88575, 13), 1151425, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2750, 531451, F27, 13) (dual of [531451, 531401, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(2749, 531442, F27, 13) (dual of [531442, 531393, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(2741, 531442, F27, 11) (dual of [531442, 531401, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(271, 9, F27, 1) (dual of [9, 8, 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([0,6]) ⊂ C([0,5]) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(2750, 531451, F27, 13) (dual of [531451, 531401, 14]-code), using
- net defined by OOA [i] based on linear OOA(2750, 88575, F27, 13, 13) (dual of [(88575, 13), 1151425, 14]-NRT-code), using
- (2, 8, 82)-net in base 27, using
(45, 45+13, 1685186)-Net over F27 — Digital
Digital (45, 58, 1685186)-net over F27, using
(45, 45+13, large)-Net in Base 27 — Upper bound on s
There is no (45, 58, large)-net in base 27, because
- 11 times m-reduction [i] would yield (45, 47, large)-net in base 27, but