Best Known (52, 52+25, s)-Nets in Base 81
(52, 52+25, 44288)-Net over F81 — Constructive and digital
Digital (52, 77, 44288)-net over F81, using
- 811 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
(52, 52+25, 265730)-Net over F81 — Digital
Digital (52, 77, 265730)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8177, 265730, F81, 2, 25) (dual of [(265730, 2), 531383, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8177, 531460, F81, 25) (dual of [531460, 531383, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(19) [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(8158, 531441, F81, 20) (dual of [531441, 531383, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(814, 19, F81, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,81)), using
- discarding factors / shortening the dual code based on linear OA(814, 81, F81, 4) (dual of [81, 77, 5]-code or 81-arc in PG(3,81)), using
- Reed–Solomon code RS(77,81) [i]
- discarding factors / shortening the dual code based on linear OA(814, 81, F81, 4) (dual of [81, 77, 5]-code or 81-arc in PG(3,81)), using
- construction X applied to Ce(24) ⊂ Ce(19) [i] based on
- OOA 2-folding [i] based on linear OA(8177, 531460, F81, 25) (dual of [531460, 531383, 26]-code), using
(52, 52+25, large)-Net in Base 81 — Upper bound on s
There is no (52, 77, large)-net in base 81, because
- 23 times m-reduction [i] would yield (52, 54, large)-net in base 81, but