Best Known (54, 54+25, s)-Nets in Base 81
(54, 54+25, 44288)-Net over F81 — Constructive and digital
Digital (54, 79, 44288)-net over F81, using
- 813 times duplication [i] based on digital (51, 76, 44288)-net over F81, using
- net defined by OOA [i] based on linear OOA(8176, 44288, F81, 25, 25) (dual of [(44288, 25), 1107124, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8176, 531457, F81, 25) (dual of [531457, 531381, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(8173, 531442, F81, 25) (dual of [531442, 531369, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(8161, 531442, F81, 21) (dual of [531442, 531381, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(813, 15, F81, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,81) or 15-cap in PG(2,81)), using
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- Reed–Solomon code RS(78,81) [i]
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- OOA 12-folding and stacking with additional row [i] based on linear OA(8176, 531457, F81, 25) (dual of [531457, 531381, 26]-code), using
- net defined by OOA [i] based on linear OOA(8176, 44288, F81, 25, 25) (dual of [(44288, 25), 1107124, 26]-NRT-code), using
(54, 54+25, 349628)-Net over F81 — Digital
Digital (54, 79, 349628)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8179, 349628, F81, 25) (dual of [349628, 349549, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8179, 531468, F81, 25) (dual of [531468, 531389, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- linear OA(8173, 531441, F81, 25) (dual of [531441, 531368, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(8152, 531441, F81, 18) (dual of [531441, 531389, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(816, 27, F81, 6) (dual of [27, 21, 7]-code or 27-arc in PG(5,81)), using
- discarding factors / shortening the dual code based on linear OA(816, 81, F81, 6) (dual of [81, 75, 7]-code or 81-arc in PG(5,81)), using
- Reed–Solomon code RS(75,81) [i]
- discarding factors / shortening the dual code based on linear OA(816, 81, F81, 6) (dual of [81, 75, 7]-code or 81-arc in PG(5,81)), using
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(8179, 531468, F81, 25) (dual of [531468, 531389, 26]-code), using
(54, 54+25, large)-Net in Base 81 — Upper bound on s
There is no (54, 79, large)-net in base 81, because
- 23 times m-reduction [i] would yield (54, 56, large)-net in base 81, but