Best Known (64, 64+31, s)-Nets in Base 27
(64, 64+31, 1314)-Net over F27 — Constructive and digital
Digital (64, 95, 1314)-net over F27, using
- net defined by OOA [i] based on linear OOA(2795, 1314, F27, 31, 31) (dual of [(1314, 31), 40639, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2795, 19711, F27, 31) (dual of [19711, 19616, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(22) [i] based on
- linear OA(2788, 19683, F27, 31) (dual of [19683, 19595, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2767, 19683, F27, 23) (dual of [19683, 19616, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(277, 28, F27, 7) (dual of [28, 21, 8]-code or 28-arc in PG(6,27)), using
- extended Reed–Solomon code RSe(21,27) [i]
- the expurgated narrow-sense BCH-code C(I) with length 28 | 272−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- construction X applied to Ce(30) ⊂ Ce(22) [i] based on
- OOA 15-folding and stacking with additional row [i] based on linear OA(2795, 19711, F27, 31) (dual of [19711, 19616, 32]-code), using
(64, 64+31, 19562)-Net over F27 — Digital
Digital (64, 95, 19562)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2795, 19562, F27, 31) (dual of [19562, 19467, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2795, 19711, F27, 31) (dual of [19711, 19616, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(22) [i] based on
- linear OA(2788, 19683, F27, 31) (dual of [19683, 19595, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2767, 19683, F27, 23) (dual of [19683, 19616, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(277, 28, F27, 7) (dual of [28, 21, 8]-code or 28-arc in PG(6,27)), using
- extended Reed–Solomon code RSe(21,27) [i]
- the expurgated narrow-sense BCH-code C(I) with length 28 | 272−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- construction X applied to Ce(30) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2795, 19711, F27, 31) (dual of [19711, 19616, 32]-code), using
(64, 64+31, large)-Net in Base 27 — Upper bound on s
There is no (64, 95, large)-net in base 27, because
- 29 times m-reduction [i] would yield (64, 66, large)-net in base 27, but