Best Known (23−5, 23, s)-Nets in Base 9
(23−5, 23, 58329)-Net over F9 — Constructive and digital
Digital (18, 23, 58329)-net over F9, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 0, 6481)-net over F9, using
- s-reduction based on digital (0, 0, s)-net over F9 with arbitrarily large s, using
- digital (0, 0, 6481)-net over F9 (see above)
- digital (0, 0, 6481)-net over F9 (see above)
- digital (0, 0, 6481)-net over F9 (see above)
- digital (0, 1, 6481)-net over F9, using
- s-reduction based on digital (0, 1, s)-net over F9 with arbitrarily large s, using
- digital (0, 1, 6481)-net over F9 (see above)
- digital (0, 1, 6481)-net over F9 (see above)
- digital (3, 5, 6481)-net over F9, using
- s-reduction based on digital (3, 5, 7381)-net over F9, using
- digital (10, 15, 6481)-net over F9, using
- net defined by OOA [i] based on linear OOA(915, 6481, F9, 5, 5) (dual of [(6481, 5), 32390, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(915, 12963, F9, 5) (dual of [12963, 12948, 6]-code), using
- 1 times code embedding in larger space [i] based on linear OA(914, 12962, F9, 5) (dual of [12962, 12948, 6]-code), using
- trace code [i] based on linear OA(817, 6481, F81, 5) (dual of [6481, 6474, 6]-code), using
- 1 times code embedding in larger space [i] based on linear OA(914, 12962, F9, 5) (dual of [12962, 12948, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(915, 12963, F9, 5) (dual of [12963, 12948, 6]-code), using
- net defined by OOA [i] based on linear OOA(915, 6481, F9, 5, 5) (dual of [(6481, 5), 32390, 6]-NRT-code), using
- digital (0, 0, 6481)-net over F9, using
(23−5, 23, 116658)-Net over F9 — Digital
Digital (18, 23, 116658)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(923, 116658, F9, 5) (dual of [116658, 116635, 6]-code), using
- generalized (u, u+v)-construction [i] based on
- linear OA(90, 12962, F9, 0) (dual of [12962, 12962, 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
- linear OA(90, 12962, F9, 0) (dual of [12962, 12962, 1]-code) (see above)
- linear OA(90, 12962, F9, 0) (dual of [12962, 12962, 1]-code) (see above)
- linear OA(90, 12962, F9, 0) (dual of [12962, 12962, 1]-code) (see above)
- linear OA(91, 12962, F9, 1) (dual of [12962, 12961, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(91, 12962, F9, 1) (dual of [12962, 12961, 2]-code) (see above)
- linear OA(91, 12962, F9, 1) (dual of [12962, 12961, 2]-code) (see above)
- linear OA(96, 12962, F9, 2) (dual of [12962, 12956, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(96, 59049, F9, 2) (dual of [59049, 59043, 3]-code), using
- an extension Ce(1) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,1], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(96, 59049, F9, 2) (dual of [59049, 59043, 3]-code), using
- linear OA(914, 12962, F9, 5) (dual of [12962, 12948, 6]-code), using
- trace code [i] based on linear OA(817, 6481, F81, 5) (dual of [6481, 6474, 6]-code), using
- linear OA(90, 12962, F9, 0) (dual of [12962, 12962, 1]-code), using
- generalized (u, u+v)-construction [i] based on
(23−5, 23, large)-Net in Base 9 — Upper bound on s
There is no (18, 23, large)-net in base 9, because
- 3 times m-reduction [i] would yield (18, 20, large)-net in base 9, but