Best Known (45, 66, s)-Nets in Base 81
(45, 66, 53146)-Net over F81 — Constructive and digital
Digital (45, 66, 53146)-net over F81, using
- net defined by OOA [i] based on linear OOA(8166, 53146, F81, 21, 21) (dual of [(53146, 21), 1116000, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8166, 531461, F81, 21) (dual of [531461, 531395, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(8166, 531465, F81, 21) (dual of [531465, 531399, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [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(8143, 531442, F81, 15) (dual of [531442, 531399, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(815, 23, F81, 5) (dual of [23, 18, 6]-code or 23-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([0,10]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8166, 531465, F81, 21) (dual of [531465, 531399, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8166, 531461, F81, 21) (dual of [531461, 531395, 22]-code), using
(45, 66, 335075)-Net over F81 — Digital
Digital (45, 66, 335075)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8166, 335075, F81, 21) (dual of [335075, 335009, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(8166, 531465, F81, 21) (dual of [531465, 531399, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [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(8143, 531442, F81, 15) (dual of [531442, 531399, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(815, 23, F81, 5) (dual of [23, 18, 6]-code or 23-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([0,10]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8166, 531465, F81, 21) (dual of [531465, 531399, 22]-code), using
(45, 66, large)-Net in Base 81 — Upper bound on s
There is no (45, 66, large)-net in base 81, because
- 19 times m-reduction [i] would yield (45, 47, large)-net in base 81, but