Best Known (48, 48+21, s)-Nets in Base 81
(48, 48+21, 53147)-Net over F81 — Constructive and digital
Digital (48, 69, 53147)-net over F81, using
- 811 times duplication [i] based on digital (47, 68, 53147)-net over F81, using
- net defined by OOA [i] based on linear OOA(8168, 53147, F81, 21, 21) (dual of [(53147, 21), 1116019, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8168, 531471, F81, 21) (dual of [531471, 531403, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(8168, 531473, F81, 21) (dual of [531473, 531405, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,6]) [i] based on
- 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(8137, 531442, F81, 13) (dual of [531442, 531405, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(817, 31, F81, 7) (dual of [31, 24, 8]-code or 31-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([0,10]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8168, 531473, F81, 21) (dual of [531473, 531405, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8168, 531471, F81, 21) (dual of [531471, 531403, 22]-code), using
- net defined by OOA [i] based on linear OOA(8168, 53147, F81, 21, 21) (dual of [(53147, 21), 1116019, 22]-NRT-code), using
(48, 48+21, 531476)-Net over F81 — Digital
Digital (48, 69, 531476)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8169, 531476, F81, 21) (dual of [531476, 531407, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(11) [i] based on
- linear OA(8161, 531441, F81, 21) (dual of [531441, 531380, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(8134, 531441, F81, 12) (dual of [531441, 531407, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(818, 35, F81, 8) (dual of [35, 27, 9]-code or 35-arc in PG(7,81)), using
- discarding factors / shortening the dual code based on linear OA(818, 81, F81, 8) (dual of [81, 73, 9]-code or 81-arc in PG(7,81)), using
- Reed–Solomon code RS(73,81) [i]
- discarding factors / shortening the dual code based on linear OA(818, 81, F81, 8) (dual of [81, 73, 9]-code or 81-arc in PG(7,81)), using
- construction X applied to Ce(20) ⊂ Ce(11) [i] based on
(48, 48+21, large)-Net in Base 81 — Upper bound on s
There is no (48, 69, large)-net in base 81, because
- 19 times m-reduction [i] would yield (48, 50, large)-net in base 81, but