Best Known (60, 60+9, s)-Nets in Base 7
(60, 60+9, 2882405)-Net over F7 — Constructive and digital
Digital (60, 69, 2882405)-net over F7, using
- 71 times duplication [i] based on digital (59, 68, 2882405)-net over F7, using
- net defined by OOA [i] based on linear OOA(768, 2882405, F7, 10, 9) (dual of [(2882405, 10), 28823982, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(768, 5764811, F7, 2, 9) (dual of [(5764811, 2), 11529554, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(768, 5764812, F7, 2, 9) (dual of [(5764812, 2), 11529556, 10]-NRT-code), using
- trace code [i] based on linear OOA(4934, 2882406, F49, 2, 9) (dual of [(2882406, 2), 5764778, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4934, 5764812, F49, 9) (dual of [5764812, 5764778, 10]-code), using
- construction X4 applied to C([0,4]) ⊂ C([0,3]) [i] based on
- linear OA(4933, 5764802, F49, 9) (dual of [5764802, 5764769, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 5764802 | 498−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(4925, 5764802, F49, 7) (dual of [5764802, 5764777, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 5764802 | 498−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(499, 10, F49, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,49)), using
- dual of repetition code with length 10 [i]
- linear OA(491, 10, F49, 1) (dual of [10, 9, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(491, 49, F49, 1) (dual of [49, 48, 2]-code), using
- Reed–Solomon code RS(48,49) [i]
- discarding factors / shortening the dual code based on linear OA(491, 49, F49, 1) (dual of [49, 48, 2]-code), using
- construction X4 applied to C([0,4]) ⊂ C([0,3]) [i] based on
- OOA 2-folding [i] based on linear OA(4934, 5764812, F49, 9) (dual of [5764812, 5764778, 10]-code), using
- trace code [i] based on linear OOA(4934, 2882406, F49, 2, 9) (dual of [(2882406, 2), 5764778, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(768, 5764812, F7, 2, 9) (dual of [(5764812, 2), 11529556, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(768, 5764811, F7, 2, 9) (dual of [(5764811, 2), 11529554, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(768, 2882405, F7, 10, 9) (dual of [(2882405, 10), 28823982, 10]-NRT-code), using
(60, 60+9, large)-Net over F7 — Digital
Digital (60, 69, large)-net over F7, using
- 75 times duplication [i] based on digital (55, 64, large)-net over F7, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(764, large, F7, 9) (dual of [large, large−64, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 20176803 | 79−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(764, large, F7, 9) (dual of [large, large−64, 10]-code), using
(60, 60+9, large)-Net in Base 7 — Upper bound on s
There is no (60, 69, large)-net in base 7, because
- 7 times m-reduction [i] would yield (60, 62, large)-net in base 7, but