Best Known (13, 13+7, s)-Nets in Base 81
(13, 13+7, 177149)-Net over F81 — Constructive and digital
Digital (13, 20, 177149)-net over F81, using
- net defined by OOA [i] based on linear OOA(8120, 177149, F81, 7, 7) (dual of [(177149, 7), 1240023, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(8120, 531448, F81, 7) (dual of [531448, 531428, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(8120, 531449, F81, 7) (dual of [531449, 531429, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [i] based on
- 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(8113, 531442, F81, 5) (dual of [531442, 531429, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (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([0,3]) ⊂ C([0,2]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8120, 531449, F81, 7) (dual of [531449, 531429, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(8120, 531448, F81, 7) (dual of [531448, 531428, 8]-code), using
(13, 13+7, 531450)-Net over F81 — Digital
Digital (13, 20, 531450)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8120, 531450, F81, 7) (dual of [531450, 531430, 8]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- 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(8113, 531442, F81, 5) (dual of [531442, 531429, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(817, 8, F81, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,81)), using
- dual of repetition code with length 8 [i]
- linear OA(811, 8, F81, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, 81, F81, 1) (dual of [81, 80, 2]-code), using
- Reed–Solomon code RS(80,81) [i]
- discarding factors / shortening the dual code based on linear OA(811, 81, F81, 1) (dual of [81, 80, 2]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
(13, 13+7, large)-Net in Base 81 — Upper bound on s
There is no (13, 20, large)-net in base 81, because
- 5 times m-reduction [i] would yield (13, 15, large)-net in base 81, but