Best Known (44, 44+9, s)-Nets in Base 27
(44, 44+9, 2107032)-Net over F27 — Constructive and digital
Digital (44, 53, 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 (32, 41, 2097150)-net over F27, using
- net defined by OOA [i] based on linear OOA(2741, 2097150, F27, 9, 9) (dual of [(2097150, 9), 18874309, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2741, 8388601, F27, 9) (dual of [8388601, 8388560, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(2741, large, F27, 9) (dual of [large, large−41, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2741, large, F27, 9) (dual of [large, large−41, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2741, 8388601, F27, 9) (dual of [8388601, 8388560, 10]-code), using
- net defined by OOA [i] based on linear OOA(2741, 2097150, F27, 9, 9) (dual of [(2097150, 9), 18874309, 10]-NRT-code), using
- digital (8, 12, 9882)-net over F27, using
(44, 44+9, large)-Net over F27 — Digital
Digital (44, 53, large)-net over F27, using
- 3 times m-reduction [i] based on digital (44, 56, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2756, large, F27, 12) (dual of [large, large−56, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2756, large, F27, 12) (dual of [large, large−56, 13]-code), using
(44, 44+9, large)-Net in Base 27 — Upper bound on s
There is no (44, 53, large)-net in base 27, because
- 7 times m-reduction [i] would yield (44, 46, large)-net in base 27, but