Best Known (40, 40+8, s)-Nets in Base 27
(40, 40+8, 2107032)-Net over F27 — Constructive and digital
Digital (40, 48, 2107032)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (8, 12, 9882)-net over F27, using
- net defined by OOA [i] based on linear OOA(2712, 9882, F27, 4, 4) (dual of [(9882, 4), 39516, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(2712, 19764, F27, 4) (dual of [19764, 19752, 5]-code), using
- generalized (u, u+v)-construction [i] based on
- linear OA(270, 732, F27, 0) (dual of [732, 732, 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
- linear OA(270, 732, F27, 0) (dual of [732, 732, 1]-code) (see above)
- linear OA(270, 732, F27, 0) (dual of [732, 732, 1]-code) (see above)
- linear OA(270, 732, F27, 0) (dual of [732, 732, 1]-code) (see above)
- linear OA(270, 732, F27, 0) (dual of [732, 732, 1]-code) (see above)
- linear OA(270, 732, F27, 0) (dual of [732, 732, 1]-code) (see above)
- linear OA(270, 732, F27, 0) (dual of [732, 732, 1]-code) (see above)
- linear OA(270, 732, F27, 0) (dual of [732, 732, 1]-code) (see above)
- linear OA(270, 732, F27, 0) (dual of [732, 732, 1]-code) (see above)
- linear OA(270, 732, F27, 0) (dual of [732, 732, 1]-code) (see above)
- linear OA(270, 732, F27, 0) (dual of [732, 732, 1]-code) (see above)
- linear OA(270, 732, F27, 0) (dual of [732, 732, 1]-code) (see above)
- linear OA(270, 732, F27, 0) (dual of [732, 732, 1]-code) (see above)
- linear OA(270, 732, F27, 0) (dual of [732, 732, 1]-code) (see above)
- linear OA(270, 732, F27, 0) (dual of [732, 732, 1]-code) (see above)
- linear OA(270, 732, F27, 0) (dual of [732, 732, 1]-code) (see above)
- linear OA(270, 732, F27, 0) (dual of [732, 732, 1]-code) (see above)
- linear OA(270, 732, F27, 0) (dual of [732, 732, 1]-code) (see above)
- linear OA(270, 732, F27, 0) (dual of [732, 732, 1]-code) (see above)
- linear OA(270, 732, F27, 0) (dual of [732, 732, 1]-code) (see above)
- linear OA(270, 732, F27, 0) (dual of [732, 732, 1]-code) (see above)
- linear OA(270, 732, F27, 0) (dual of [732, 732, 1]-code) (see above)
- linear OA(270, 732, F27, 0) (dual of [732, 732, 1]-code) (see above)
- linear OA(271, 732, F27, 1) (dual of [732, 731, 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
- linear OA(271, 732, F27, 1) (dual of [732, 731, 2]-code) (see above)
- linear OA(273, 732, F27, 2) (dual of [732, 729, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(273, 757, F27, 2) (dual of [757, 754, 3]-code), using
- Hamming code H(3,27) [i]
- discarding factors / shortening the dual code based on linear OA(273, 757, F27, 2) (dual of [757, 754, 3]-code), using
- linear OA(277, 732, F27, 4) (dual of [732, 725, 5]-code), using
- construction XX applied to C1 = C([727,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([727,2]) [i] based on
- linear OA(275, 728, F27, 3) (dual of [728, 723, 4]-code or 728-cap in PG(4,27)), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,1}, and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(275, 728, F27, 3) (dual of [728, 723, 4]-code or 728-cap in PG(4,27)), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(277, 728, F27, 4) (dual of [728, 721, 5]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,1,2}, and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(273, 728, F27, 2) (dual of [728, 725, 3]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(270, 2, F27, 0) (dual of [2, 2, 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 (see above)
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([727,1]), C2 = C([0,2]), C3 = C1 + C2 = C([0,1]), and C∩ = C1 ∩ C2 = C([727,2]) [i] based on
- linear OA(270, 732, F27, 0) (dual of [732, 732, 1]-code), using
- generalized (u, u+v)-construction [i] based on
- OA 2-folding and stacking [i] based on linear OA(2712, 19764, F27, 4) (dual of [19764, 19752, 5]-code), using
- net defined by OOA [i] based on linear OOA(2712, 9882, F27, 4, 4) (dual of [(9882, 4), 39516, 5]-NRT-code), 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
- digital (8, 12, 9882)-net over F27, using
(40, 40+8, large)-Net over F27 — Digital
Digital (40, 48, large)-net over F27, using
- 3 times m-reduction [i] based on digital (40, 51, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] 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]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2751, large, F27, 11) (dual of [large, large−51, 12]-code), using
(40, 40+8, large)-Net in Base 27 — Upper bound on s
There is no (40, 48, large)-net in base 27, because
- 6 times m-reduction [i] would yield (40, 42, large)-net in base 27, but