Best Known (80, 80+17, s)-Nets in Base 27
(80, 80+17, 1048758)-Net over F27 — Constructive and digital
Digital (80, 97, 1048758)-net over F27, using
- 271 times duplication [i] based on digital (79, 96, 1048758)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 15, 183)-net over F27, using
- net defined by OOA [i] based on linear OOA(2715, 183, F27, 8, 8) (dual of [(183, 8), 1449, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(2715, 732, F27, 8) (dual of [732, 717, 9]-code), using
- construction XX applied to C1 = C([727,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([727,6]) [i] based on
- linear OA(2713, 728, F27, 7) (dual of [728, 715, 8]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,5}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(2713, 728, F27, 7) (dual of [728, 715, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(2715, 728, F27, 8) (dual of [728, 713, 9]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,6}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(2711, 728, F27, 6) (dual of [728, 717, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [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, using
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([727,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([727,6]) [i] based on
- OA 4-folding and stacking [i] based on linear OA(2715, 732, F27, 8) (dual of [732, 717, 9]-code), using
- net defined by OOA [i] based on linear OOA(2715, 183, F27, 8, 8) (dual of [(183, 8), 1449, 9]-NRT-code), using
- digital (64, 81, 1048575)-net over F27, using
- net defined by OOA [i] based on linear OOA(2781, 1048575, F27, 17, 17) (dual of [(1048575, 17), 17825694, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2781, 8388601, F27, 17) (dual of [8388601, 8388520, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2781, large, F27, 17) (dual of [large, large−81, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2781, large, F27, 17) (dual of [large, large−81, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2781, 8388601, F27, 17) (dual of [8388601, 8388520, 18]-code), using
- net defined by OOA [i] based on linear OOA(2781, 1048575, F27, 17, 17) (dual of [(1048575, 17), 17825694, 18]-NRT-code), using
- digital (7, 15, 183)-net over F27, using
- (u, u+v)-construction [i] based on
(80, 80+17, large)-Net over F27 — Digital
Digital (80, 97, large)-net over F27, using
- 4 times m-reduction [i] based on digital (80, 101, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(27101, large, F27, 21) (dual of [large, large−101, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(27101, large, F27, 21) (dual of [large, large−101, 22]-code), using
(80, 80+17, large)-Net in Base 27 — Upper bound on s
There is no (80, 97, large)-net in base 27, because
- 15 times m-reduction [i] would yield (80, 82, large)-net in base 27, but