Best Known (53, 53+7, s)-Nets in Base 9
(53, 53+7, 6137045)-Net over F9 — Constructive and digital
Digital (53, 60, 6137045)-net over F9, using
- net defined by OOA [i] based on linear OOA(960, 6137045, F9, 9, 7) (dual of [(6137045, 9), 55233345, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(960, 6137046, F9, 3, 7) (dual of [(6137046, 3), 18411078, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(910, 544644, F9, 3, 3) (dual of [(544644, 3), 1633922, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(910, 544644, F9, 2, 3) (dual of [(544644, 2), 1089278, 4]-NRT-code), using
- linear OOA(950, 5592402, F9, 3, 7) (dual of [(5592402, 3), 16777156, 8]-NRT-code), using
- trace code [i] based on linear OOA(8125, 2796201, F81, 3, 7) (dual of [(2796201, 3), 8388578, 8]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8125, large, F81, 7) (dual of [large, large−25, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- OOA 3-folding [i] based on linear OA(8125, large, F81, 7) (dual of [large, large−25, 8]-code), using
- trace code [i] based on linear OOA(8125, 2796201, F81, 3, 7) (dual of [(2796201, 3), 8388578, 8]-NRT-code), using
- linear OOA(910, 544644, F9, 3, 3) (dual of [(544644, 3), 1633922, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(960, 6137046, F9, 3, 7) (dual of [(6137046, 3), 18411078, 8]-NRT-code), using
(53, 53+7, large)-Net over F9 — Digital
Digital (53, 60, large)-net over F9, using
- t-expansion [i] based on digital (52, 60, large)-net over F9, using
- 1 times m-reduction [i] based on digital (52, 61, large)-net over F9, using
(53, 53+7, large)-Net in Base 9 — Upper bound on s
There is no (53, 60, large)-net in base 9, because
- 5 times m-reduction [i] would yield (53, 55, large)-net in base 9, but