Best Known (50, 50+11, s)-Nets in Base 27
(50, 50+11, 1678476)-Net over F27 — Constructive and digital
Digital (50, 61, 1678476)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (5, 10, 756)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 0, 28)-net over F27, using
- s-reduction based on digital (0, 0, s)-net over F27 with arbitrarily large s, using
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- 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, 1, 28)-net over F27 (see above)
- digital (0, 2, 28)-net over F27, using
- digital (0, 5, 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, 0, 28)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (40, 51, 1677720)-net over F27, using
- net defined by OOA [i] based on linear OOA(2751, 1677720, F27, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2751, 8388601, F27, 11) (dual of [8388601, 8388550, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(2751, large, F27, 11) (dual of [large, large−51, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2751, large, F27, 11) (dual of [large, large−51, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2751, 8388601, F27, 11) (dual of [8388601, 8388550, 12]-code), using
- net defined by OOA [i] based on linear OOA(2751, 1677720, F27, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- digital (5, 10, 756)-net over F27, using
(50, 50+11, 1680960)-Net in Base 27 — Constructive
(50, 61, 1680960)-net in base 27, using
- (u, u+v)-construction [i] based on
- (5, 10, 3240)-net in base 27, using
- net defined by OOA [i] based on OOA(2710, 3240, S27, 5, 5), using
- OOA 2-folding and stacking with additional row [i] based on OA(2710, 6481, S27, 5), using
- discarding parts of the base [i] based on linear OA(817, 6481, F81, 5) (dual of [6481, 6474, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on OA(2710, 6481, S27, 5), using
- net defined by OOA [i] based on OOA(2710, 3240, S27, 5, 5), using
- digital (40, 51, 1677720)-net over F27, using
- net defined by OOA [i] based on linear OOA(2751, 1677720, F27, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2751, 8388601, F27, 11) (dual of [8388601, 8388550, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(2751, large, F27, 11) (dual of [large, large−51, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2751, large, F27, 11) (dual of [large, large−51, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2751, 8388601, F27, 11) (dual of [8388601, 8388550, 12]-code), using
- net defined by OOA [i] based on linear OOA(2751, 1677720, F27, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- (5, 10, 3240)-net in base 27, using
(50, 50+11, large)-Net over F27 — Digital
Digital (50, 61, large)-net over F27, using
- t-expansion [i] based on digital (48, 61, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2761, large, F27, 13) (dual of [large, large−61, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2761, large, F27, 13) (dual of [large, large−61, 14]-code), using
(50, 50+11, large)-Net in Base 27 — Upper bound on s
There is no (50, 61, large)-net in base 27, because
- 9 times m-reduction [i] would yield (50, 52, large)-net in base 27, but