Best Known (31, 57, s)-Nets in Base 81
(31, 57, 506)-Net over F81 — Constructive and digital
Digital (31, 57, 506)-net over F81, using
- 1 times m-reduction [i] based on digital (31, 58, 506)-net over F81, using
- net defined by OOA [i] based on linear OOA(8158, 506, F81, 27, 27) (dual of [(506, 27), 13604, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(8158, 6579, F81, 27) (dual of [6579, 6521, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,10]) [i] based on
- linear OA(8153, 6562, F81, 27) (dual of [6562, 6509, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(8141, 6562, F81, 21) (dual of [6562, 6521, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (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([0,13]) ⊂ C([0,10]) [i] based on
- OOA 13-folding and stacking with additional row [i] based on linear OA(8158, 6579, F81, 27) (dual of [6579, 6521, 28]-code), using
- net defined by OOA [i] based on linear OOA(8158, 506, F81, 27, 27) (dual of [(506, 27), 13604, 28]-NRT-code), using
(31, 57, 3467)-Net over F81 — Digital
Digital (31, 57, 3467)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8157, 3467, F81, 26) (dual of [3467, 3410, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(8157, 6581, F81, 26) (dual of [6581, 6524, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(18) [i] based on
- linear OA(8151, 6561, F81, 26) (dual of [6561, 6510, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(8137, 6561, F81, 19) (dual of [6561, 6524, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(816, 20, F81, 6) (dual of [20, 14, 7]-code or 20-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(25) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(8157, 6581, F81, 26) (dual of [6581, 6524, 27]-code), using
(31, 57, large)-Net in Base 81 — Upper bound on s
There is no (31, 57, large)-net in base 81, because
- 24 times m-reduction [i] would yield (31, 33, large)-net in base 81, but