Best Known (82−27, 82, s)-Nets in Base 81
(82−27, 82, 40881)-Net over F81 — Constructive and digital
Digital (55, 82, 40881)-net over F81, using
- net defined by OOA [i] based on linear OOA(8182, 40881, F81, 27, 27) (dual of [(40881, 27), 1103705, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(8182, 531454, F81, 27) (dual of [531454, 531372, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(8182, 531457, F81, 27) (dual of [531457, 531375, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [i] based on
- linear OA(8179, 531442, F81, 27) (dual of [531442, 531363, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(8167, 531442, F81, 23) (dual of [531442, 531375, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(813, 15, F81, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,81) or 15-cap in PG(2,81)), using
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- Reed–Solomon code RS(78,81) [i]
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8182, 531457, F81, 27) (dual of [531457, 531375, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(8182, 531454, F81, 27) (dual of [531454, 531372, 28]-code), using
(82−27, 82, 265728)-Net over F81 — Digital
Digital (55, 82, 265728)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8182, 265728, F81, 2, 27) (dual of [(265728, 2), 531374, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8182, 531456, F81, 27) (dual of [531456, 531374, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(8182, 531457, F81, 27) (dual of [531457, 531375, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [i] based on
- linear OA(8179, 531442, F81, 27) (dual of [531442, 531363, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(8167, 531442, F81, 23) (dual of [531442, 531375, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(813, 15, F81, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,81) or 15-cap in PG(2,81)), using
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- Reed–Solomon code RS(78,81) [i]
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8182, 531457, F81, 27) (dual of [531457, 531375, 28]-code), using
- OOA 2-folding [i] based on linear OA(8182, 531456, F81, 27) (dual of [531456, 531374, 28]-code), using
(82−27, 82, large)-Net in Base 81 — Upper bound on s
There is no (55, 82, large)-net in base 81, because
- 25 times m-reduction [i] would yield (55, 57, large)-net in base 81, but