Best Known (17, 26, s)-Nets in Base 81
(17, 26, 132862)-Net over F81 — Constructive and digital
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
(17, 26, 276594)-Net over F81 — Digital
Digital (17, 26, 276594)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8126, 276594, F81, 9) (dual of [276594, 276568, 10]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(8126, 531449, F81, 9) (dual of [531449, 531423, 10]-code), using
(17, 26, large)-Net in Base 81 — Upper bound on s
There is no (17, 26, large)-net in base 81, because
- 7 times m-reduction [i] would yield (17, 19, large)-net in base 81, but