Best Known (62−19, 62, s)-Nets in Base 81
(62−19, 62, 59052)-Net over F81 — Constructive and digital
Digital (43, 62, 59052)-net over F81, using
- net defined by OOA [i] based on linear OOA(8162, 59052, F81, 19, 19) (dual of [(59052, 19), 1121926, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(8162, 531469, F81, 19) (dual of [531469, 531407, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(8162, 531473, F81, 19) (dual of [531473, 531411, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,5]) [i] based on
- linear OA(8155, 531442, F81, 19) (dual of [531442, 531387, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(8131, 531442, F81, 11) (dual of [531442, 531411, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,9]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8162, 531473, F81, 19) (dual of [531473, 531411, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(8162, 531469, F81, 19) (dual of [531469, 531407, 20]-code), using
(62−19, 62, 531473)-Net over F81 — Digital
Digital (43, 62, 531473)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8162, 531473, F81, 19) (dual of [531473, 531411, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,5]) [i] based on
- linear OA(8155, 531442, F81, 19) (dual of [531442, 531387, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(8131, 531442, F81, 11) (dual of [531442, 531411, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (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([0,9]) ⊂ C([0,5]) [i] based on
(62−19, 62, large)-Net in Base 81 — Upper bound on s
There is no (43, 62, large)-net in base 81, because
- 17 times m-reduction [i] would yield (43, 45, large)-net in base 81, but