Best Known (95, 95+14, s)-Nets in Base 27
(95, 95+14, 2406624)-Net over F27 — Constructive and digital
Digital (95, 109, 2406624)-net over F27, using
- generalized (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 (24, 31, 1198371)-net over F27, using
- s-reduction based on digital (24, 31, 2796200)-net over F27, using
- net defined by OOA [i] based on linear OOA(2731, 2796200, F27, 7, 7) (dual of [(2796200, 7), 19573369, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2731, 8388601, F27, 7) (dual of [8388601, 8388570, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(2731, large, F27, 7) (dual of [large, large−31, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2731, large, F27, 7) (dual of [large, large−31, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2731, 8388601, F27, 7) (dual of [8388601, 8388570, 8]-code), using
- net defined by OOA [i] based on linear OOA(2731, 2796200, F27, 7, 7) (dual of [(2796200, 7), 19573369, 8]-NRT-code), using
- s-reduction based on digital (24, 31, 2796200)-net over F27, using
- digital (52, 66, 1198371)-net over F27, using
- net defined by OOA [i] based on linear OOA(2766, 1198371, F27, 14, 14) (dual of [(1198371, 14), 16777128, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(2766, 8388597, F27, 14) (dual of [8388597, 8388531, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(2766, large, F27, 14) (dual of [large, large−66, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(2766, large, F27, 14) (dual of [large, large−66, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(2766, 8388597, F27, 14) (dual of [8388597, 8388531, 15]-code), using
- net defined by OOA [i] based on linear OOA(2766, 1198371, F27, 14, 14) (dual of [(1198371, 14), 16777128, 15]-NRT-code), using
- digital (8, 12, 9882)-net over F27, using
(95, 95+14, large)-Net over F27 — Digital
Digital (95, 109, large)-net over F27, using
- 273 times duplication [i] based on digital (92, 106, large)-net over F27, using
- t-expansion [i] based on digital (84, 106, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(27106, large, F27, 22) (dual of [large, large−106, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(27106, large, F27, 22) (dual of [large, large−106, 23]-code), using
- t-expansion [i] based on digital (84, 106, large)-net over F27, using
(95, 95+14, large)-Net in Base 27 — Upper bound on s
There is no (95, 109, large)-net in base 27, because
- 12 times m-reduction [i] would yield (95, 97, large)-net in base 27, but