Best Known (36, 36+8, s)-Nets in Base 27
(36, 36+8, 2097906)-Net over F27 — Constructive and digital
Digital (36, 44, 2097906)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (4, 8, 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, 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, 2, 28)-net over F27, using
- digital (0, 4, 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 (28, 36, 2097150)-net over F27, using
- net defined by OOA [i] based on linear OOA(2736, 2097150, F27, 8, 8) (dual of [(2097150, 8), 16777164, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(2736, 8388600, F27, 8) (dual of [8388600, 8388564, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(2736, large, F27, 8) (dual of [large, large−36, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(2736, large, F27, 8) (dual of [large, large−36, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(2736, 8388600, F27, 8) (dual of [8388600, 8388564, 9]-code), using
- net defined by OOA [i] based on linear OOA(2736, 2097150, F27, 8, 8) (dual of [(2097150, 8), 16777164, 9]-NRT-code), using
- digital (4, 8, 756)-net over F27, using
(36, 36+8, 2100390)-Net in Base 27 — Constructive
(36, 44, 2100390)-net in base 27, using
- (u, u+v)-construction [i] based on
- (4, 8, 3240)-net in base 27, using
- base change [i] based on digital (2, 6, 3240)-net over F81, using
- net defined by OOA [i] based on linear OOA(816, 3240, F81, 4, 4) (dual of [(3240, 4), 12954, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(816, 6480, F81, 4) (dual of [6480, 6474, 5]-code), using
- 1 times truncation [i] based on linear OA(817, 6481, F81, 5) (dual of [6481, 6474, 6]-code), using
- OA 2-folding and stacking [i] based on linear OA(816, 6480, F81, 4) (dual of [6480, 6474, 5]-code), using
- net defined by OOA [i] based on linear OOA(816, 3240, F81, 4, 4) (dual of [(3240, 4), 12954, 5]-NRT-code), using
- base change [i] based on digital (2, 6, 3240)-net over F81, using
- digital (28, 36, 2097150)-net over F27, using
- net defined by OOA [i] based on linear OOA(2736, 2097150, F27, 8, 8) (dual of [(2097150, 8), 16777164, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(2736, 8388600, F27, 8) (dual of [8388600, 8388564, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(2736, large, F27, 8) (dual of [large, large−36, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(2736, large, F27, 8) (dual of [large, large−36, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(2736, 8388600, F27, 8) (dual of [8388600, 8388564, 9]-code), using
- net defined by OOA [i] based on linear OOA(2736, 2097150, F27, 8, 8) (dual of [(2097150, 8), 16777164, 9]-NRT-code), using
- (4, 8, 3240)-net in base 27, using
(36, 36+8, large)-Net over F27 — Digital
Digital (36, 44, large)-net over F27, using
- 2 times m-reduction [i] based on digital (36, 46, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2746, large, F27, 10) (dual of [large, large−46, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2746, large, F27, 10) (dual of [large, large−46, 11]-code), using
(36, 36+8, large)-Net in Base 27 — Upper bound on s
There is no (36, 44, large)-net in base 27, because
- 6 times m-reduction [i] would yield (36, 38, large)-net in base 27, but