Best Known (31, 31+25, s)-Nets in Base 81
(31, 31+25, 548)-Net over F81 — Constructive and digital
Digital (31, 56, 548)-net over F81, using
- 812 times duplication [i] based on digital (29, 54, 548)-net over F81, using
- net defined by OOA [i] based on linear OOA(8154, 548, F81, 25, 25) (dual of [(548, 25), 13646, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8154, 6577, F81, 25) (dual of [6577, 6523, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8154, 6579, F81, 25) (dual of [6579, 6525, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(8149, 6562, F81, 25) (dual of [6562, 6513, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(8137, 6562, F81, 19) (dual of [6562, 6525, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(815, 17, F81, 5) (dual of [17, 12, 6]-code or 17-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([0,12]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8154, 6579, F81, 25) (dual of [6579, 6525, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8154, 6577, F81, 25) (dual of [6577, 6523, 26]-code), using
- net defined by OOA [i] based on linear OOA(8154, 548, F81, 25, 25) (dual of [(548, 25), 13646, 26]-NRT-code), using
(31, 31+25, 4306)-Net over F81 — Digital
Digital (31, 56, 4306)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8156, 4306, F81, 25) (dual of [4306, 4250, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8156, 6585, F81, 25) (dual of [6585, 6529, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- linear OA(8149, 6562, F81, 25) (dual of [6562, 6513, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(8133, 6562, F81, 17) (dual of [6562, 6529, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(817, 23, F81, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,81)), using
- discarding factors / shortening the dual code based on linear OA(817, 81, F81, 7) (dual of [81, 74, 8]-code or 81-arc in PG(6,81)), using
- Reed–Solomon code RS(74,81) [i]
- discarding factors / shortening the dual code based on linear OA(817, 81, F81, 7) (dual of [81, 74, 8]-code or 81-arc in PG(6,81)), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8156, 6585, F81, 25) (dual of [6585, 6529, 26]-code), using
(31, 31+25, large)-Net in Base 81 — Upper bound on s
There is no (31, 56, large)-net in base 81, because
- 23 times m-reduction [i] would yield (31, 33, large)-net in base 81, but