Best Known (18, 27, s)-Nets in Base 81
(18, 27, 132862)-Net over F81 — Constructive and digital
Digital (18, 27, 132862)-net over F81, using
- 811 times duplication [i] based on digital (17, 26, 132862)-net over F81, using
- net defined by OOA [i] based on linear OOA(8126, 132862, F81, 9, 9) (dual of [(132862, 9), 1195732, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(8126, 531449, F81, 9) (dual of [531449, 531423, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(8125, 531442, F81, 9) (dual of [531442, 531417, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(8119, 531442, F81, 7) (dual of [531442, 531423, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(811, 7, F81, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,4]) ⊂ C([0,3]) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(8126, 531449, F81, 9) (dual of [531449, 531423, 10]-code), using
- net defined by OOA [i] based on linear OOA(8126, 132862, F81, 9, 9) (dual of [(132862, 9), 1195732, 10]-NRT-code), using
(18, 27, 518185)-Net over F81 — Digital
Digital (18, 27, 518185)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8127, 518185, F81, 9) (dual of [518185, 518158, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(8127, 531452, F81, 9) (dual of [531452, 531425, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(8125, 531441, F81, 9) (dual of [531441, 531416, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(8116, 531441, F81, 6) (dual of [531441, 531425, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(812, 11, F81, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,81)), using
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- Reed–Solomon code RS(79,81) [i]
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(8127, 531452, F81, 9) (dual of [531452, 531425, 10]-code), using
(18, 27, large)-Net in Base 81 — Upper bound on s
There is no (18, 27, large)-net in base 81, because
- 7 times m-reduction [i] would yield (18, 20, large)-net in base 81, but