Best Known (69, 69+23, s)-Nets in Base 9
(69, 69+23, 1193)-Net over F9 — Constructive and digital
Digital (69, 92, 1193)-net over F9, using
- 92 times duplication [i] based on digital (67, 90, 1193)-net over F9, using
- net defined by OOA [i] based on linear OOA(990, 1193, F9, 23, 23) (dual of [(1193, 23), 27349, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(990, 13124, F9, 23) (dual of [13124, 13034, 24]-code), using
- trace code [i] based on linear OA(8145, 6562, F81, 23) (dual of [6562, 6517, 24]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- trace code [i] based on linear OA(8145, 6562, F81, 23) (dual of [6562, 6517, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(990, 13124, F9, 23) (dual of [13124, 13034, 24]-code), using
- net defined by OOA [i] based on linear OOA(990, 1193, F9, 23, 23) (dual of [(1193, 23), 27349, 24]-NRT-code), using
(69, 69+23, 13134)-Net over F9 — Digital
Digital (69, 92, 13134)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(992, 13134, F9, 23) (dual of [13134, 13042, 24]-code), using
- trace code [i] based on linear OA(8146, 6567, F81, 23) (dual of [6567, 6521, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(8145, 6562, F81, 23) (dual of [6562, 6517, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(8141, 6562, F81, 21) (dual of [6562, 6521, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 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([0,11]) ⊂ C([0,10]) [i] based on
- trace code [i] based on linear OA(8146, 6567, F81, 23) (dual of [6567, 6521, 24]-code), using
(69, 69+23, large)-Net in Base 9 — Upper bound on s
There is no (69, 92, large)-net in base 9, because
- 21 times m-reduction [i] would yield (69, 71, large)-net in base 9, but