Best Known (69, 77, s)-Nets in Base 9
(69, 77, 4460023)-Net over F9 — Constructive and digital
Digital (69, 77, 4460023)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (15, 19, 265723)-net over F9, using
- net defined by OOA [i] based on linear OOA(919, 265723, F9, 4, 4) (dual of [(265723, 4), 1062873, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(919, 265723, F9, 3, 4) (dual of [(265723, 3), 797150, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(919, 531446, F9, 4) (dual of [531446, 531427, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(919, 531447, F9, 4) (dual of [531447, 531428, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(919, 531441, F9, 4) (dual of [531441, 531422, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(913, 531441, F9, 3) (dual of [531441, 531428, 4]-code or 531441-cap in PG(12,9)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(90, 6, F9, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(919, 531447, F9, 4) (dual of [531447, 531428, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(919, 531446, F9, 4) (dual of [531446, 531427, 5]-code), using
- appending kth column [i] based on linear OOA(919, 265723, F9, 3, 4) (dual of [(265723, 3), 797150, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(919, 265723, F9, 4, 4) (dual of [(265723, 4), 1062873, 5]-NRT-code), using
- digital (50, 58, 4194300)-net over F9, using
- net defined by OOA [i] based on linear OOA(958, 4194300, F9, 10, 8) (dual of [(4194300, 10), 41942942, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(958, 8388601, F9, 2, 8) (dual of [(8388601, 2), 16777144, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(958, 8388602, F9, 2, 8) (dual of [(8388602, 2), 16777146, 9]-NRT-code), using
- trace code [i] based on linear OOA(8129, 4194301, F81, 2, 8) (dual of [(4194301, 2), 8388573, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8129, 8388602, F81, 8) (dual of [8388602, 8388573, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(8129, large, F81, 8) (dual of [large, large−29, 9]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(8129, large, F81, 8) (dual of [large, large−29, 9]-code), using
- OOA 2-folding [i] based on linear OA(8129, 8388602, F81, 8) (dual of [8388602, 8388573, 9]-code), using
- trace code [i] based on linear OOA(8129, 4194301, F81, 2, 8) (dual of [(4194301, 2), 8388573, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(958, 8388602, F9, 2, 8) (dual of [(8388602, 2), 16777146, 9]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(958, 8388601, F9, 2, 8) (dual of [(8388601, 2), 16777144, 9]-NRT-code), using
- net defined by OOA [i] based on linear OOA(958, 4194300, F9, 10, 8) (dual of [(4194300, 10), 41942942, 9]-NRT-code), using
- digital (15, 19, 265723)-net over F9, using
(69, 77, large)-Net over F9 — Digital
Digital (69, 77, large)-net over F9, using
- 4 times m-reduction [i] based on digital (69, 81, large)-net over F9, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(981, large, F9, 12) (dual of [large, large−81, 13]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 98−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(981, large, F9, 12) (dual of [large, large−81, 13]-code), using
(69, 77, large)-Net in Base 9 — Upper bound on s
There is no (69, 77, large)-net in base 9, because
- 6 times m-reduction [i] would yield (69, 71, large)-net in base 9, but