Best Known (58, 58+17, s)-Nets in Base 9
(58, 58+17, 1656)-Net over F9 — Constructive and digital
Digital (58, 75, 1656)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (1, 9, 16)-net over F9, using
- net from sequence [i] based on digital (1, 15)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 1 and N(F) ≥ 16, using
- net from sequence [i] based on digital (1, 15)-sequence over F9, using
- digital (49, 66, 1640)-net over F9, using
- net defined by OOA [i] based on linear OOA(966, 1640, F9, 17, 17) (dual of [(1640, 17), 27814, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(966, 13121, F9, 17) (dual of [13121, 13055, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(966, 13124, F9, 17) (dual of [13124, 13058, 18]-code), using
- trace code [i] based on linear OA(8133, 6562, F81, 17) (dual of [6562, 6529, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- trace code [i] based on linear OA(8133, 6562, F81, 17) (dual of [6562, 6529, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(966, 13124, F9, 17) (dual of [13124, 13058, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(966, 13121, F9, 17) (dual of [13121, 13055, 18]-code), using
- net defined by OOA [i] based on linear OOA(966, 1640, F9, 17, 17) (dual of [(1640, 17), 27814, 18]-NRT-code), using
- digital (1, 9, 16)-net over F9, using
(58, 58+17, 2461)-Net in Base 9 — Constructive
(58, 75, 2461)-net in base 9, using
- base change [i] based on digital (33, 50, 2461)-net over F27, using
- net defined by OOA [i] based on linear OOA(2750, 2461, F27, 17, 17) (dual of [(2461, 17), 41787, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2750, 19689, F27, 17) (dual of [19689, 19639, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2750, 19691, F27, 17) (dual of [19691, 19641, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- linear OA(2749, 19684, F27, 17) (dual of [19684, 19635, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(2743, 19684, F27, 15) (dual of [19684, 19641, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(271, 7, F27, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, s, F27, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,8]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2750, 19691, F27, 17) (dual of [19691, 19641, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2750, 19689, F27, 17) (dual of [19689, 19639, 18]-code), using
- net defined by OOA [i] based on linear OOA(2750, 2461, F27, 17, 17) (dual of [(2461, 17), 41787, 18]-NRT-code), using
(58, 58+17, 25269)-Net over F9 — Digital
Digital (58, 75, 25269)-net over F9, using
(58, 58+17, large)-Net in Base 9 — Upper bound on s
There is no (58, 75, large)-net in base 9, because
- 15 times m-reduction [i] would yield (58, 60, large)-net in base 9, but