Best Known (54−17, 54, s)-Nets in Base 81
(54−17, 54, 66433)-Net over F81 — Constructive and digital
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
(54−17, 54, 444615)-Net over F81 — Digital
Digital (37, 54, 444615)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8154, 444615, F81, 17) (dual of [444615, 444561, 18]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(8154, 531465, F81, 17) (dual of [531465, 531411, 18]-code), using
(54−17, 54, large)-Net in Base 81 — Upper bound on s
There is no (37, 54, large)-net in base 81, because
- 15 times m-reduction [i] would yield (37, 39, large)-net in base 81, but