Best Known (55−17, 55, s)-Nets in Base 81
(55−17, 55, 66433)-Net over F81 — Constructive and digital
Digital (38, 55, 66433)-net over F81, using
- 811 times duplication [i] based on digital (37, 54, 66433)-net over F81, using
- net defined by OOA [i] based on linear OOA(8154, 66433, F81, 17, 17) (dual of [(66433, 17), 1129307, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(8154, 531465, F81, 17) (dual of [531465, 531411, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- linear OA(8149, 531442, F81, 17) (dual of [531442, 531393, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(8131, 531442, F81, 11) (dual of [531442, 531411, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(815, 23, F81, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,81)), using
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- Reed–Solomon code RS(76,81) [i]
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- OOA 8-folding and stacking with additional row [i] based on linear OA(8154, 531465, F81, 17) (dual of [531465, 531411, 18]-code), using
- net defined by OOA [i] based on linear OOA(8154, 66433, F81, 17, 17) (dual of [(66433, 17), 1129307, 18]-NRT-code), using
(55−17, 55, 531468)-Net over F81 — Digital
Digital (38, 55, 531468)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8155, 531468, F81, 17) (dual of [531468, 531413, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(9) [i] based on
- linear OA(8149, 531441, F81, 17) (dual of [531441, 531392, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(8128, 531441, F81, 10) (dual of [531441, 531413, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(16) ⊂ Ce(9) [i] based on
(55−17, 55, large)-Net in Base 81 — Upper bound on s
There is no (38, 55, large)-net in base 81, because
- 15 times m-reduction [i] would yield (38, 40, large)-net in base 81, but