Best Known (109−11, 109, s)-Nets in Base 9
(109−11, 109, 3886884)-Net over F9 — Constructive and digital
Digital (98, 109, 3886884)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (22, 27, 531444)-net over F9, using
- net defined by OOA [i] based on linear OOA(927, 531444, F9, 5, 5) (dual of [(531444, 5), 2657193, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(927, 1062889, F9, 5) (dual of [1062889, 1062862, 6]-code), using
- 1 times code embedding in larger space [i] based on linear OA(926, 1062888, F9, 5) (dual of [1062888, 1062862, 6]-code), using
- trace code [i] based on linear OA(8113, 531444, F81, 5) (dual of [531444, 531431, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(8113, 531441, F81, 5) (dual of [531441, 531428, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(8110, 531441, F81, 4) (dual of [531441, 531431, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(810, 3, F81, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- trace code [i] based on linear OA(8113, 531444, F81, 5) (dual of [531444, 531431, 6]-code), using
- 1 times code embedding in larger space [i] based on linear OA(926, 1062888, F9, 5) (dual of [1062888, 1062862, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(927, 1062889, F9, 5) (dual of [1062889, 1062862, 6]-code), using
- net defined by OOA [i] based on linear OOA(927, 531444, F9, 5, 5) (dual of [(531444, 5), 2657193, 6]-NRT-code), using
- digital (71, 82, 3355440)-net over F9, using
- trace code for nets [i] based on digital (30, 41, 1677720)-net over F81, using
- net defined by OOA [i] based on linear OOA(8141, 1677720, F81, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(8141, 8388601, F81, 11) (dual of [8388601, 8388560, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(8141, large, F81, 11) (dual of [large, large−41, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8141, large, F81, 11) (dual of [large, large−41, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(8141, 8388601, F81, 11) (dual of [8388601, 8388560, 12]-code), using
- net defined by OOA [i] based on linear OOA(8141, 1677720, F81, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- trace code for nets [i] based on digital (30, 41, 1677720)-net over F81, using
- digital (22, 27, 531444)-net over F9, using
(109−11, 109, large)-Net over F9 — Digital
Digital (98, 109, large)-net over F9, using
- t-expansion [i] based on digital (95, 109, large)-net over F9, using
- 2 times m-reduction [i] based on digital (95, 111, large)-net over F9, using
(109−11, 109, large)-Net in Base 9 — Upper bound on s
There is no (98, 109, large)-net in base 9, because
- 9 times m-reduction [i] would yield (98, 100, large)-net in base 9, but