Best Known (123, 148, s)-Nets in Base 9
(123, 148, 88574)-Net over F9 — Constructive and digital
Digital (123, 148, 88574)-net over F9, using
- 91 times duplication [i] based on digital (122, 147, 88574)-net over F9, using
- net defined by OOA [i] based on linear OOA(9147, 88574, F9, 25, 25) (dual of [(88574, 25), 2214203, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(9147, 1062889, F9, 25) (dual of [1062889, 1062742, 26]-code), using
- 1 times code embedding in larger space [i] based on linear OA(9146, 1062888, F9, 25) (dual of [1062888, 1062742, 26]-code), using
- trace code [i] based on linear OA(8173, 531444, F81, 25) (dual of [531444, 531371, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(23) [i] based on
- linear OA(8173, 531441, F81, 25) (dual of [531441, 531368, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(8170, 531441, F81, 24) (dual of [531441, 531371, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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(24) ⊂ Ce(23) [i] based on
- trace code [i] based on linear OA(8173, 531444, F81, 25) (dual of [531444, 531371, 26]-code), using
- 1 times code embedding in larger space [i] based on linear OA(9146, 1062888, F9, 25) (dual of [1062888, 1062742, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(9147, 1062889, F9, 25) (dual of [1062889, 1062742, 26]-code), using
- net defined by OOA [i] based on linear OOA(9147, 88574, F9, 25, 25) (dual of [(88574, 25), 2214203, 26]-NRT-code), using
(123, 148, 1062898)-Net over F9 — Digital
Digital (123, 148, 1062898)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9148, 1062898, F9, 25) (dual of [1062898, 1062750, 26]-code), using
- trace code [i] based on linear OA(8174, 531449, F81, 25) (dual of [531449, 531375, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- linear OA(8173, 531442, F81, 25) (dual of [531442, 531369, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(8167, 531442, F81, 23) (dual of [531442, 531375, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(811, 7, F81, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- trace code [i] based on linear OA(8174, 531449, F81, 25) (dual of [531449, 531375, 26]-code), using
(123, 148, large)-Net in Base 9 — Upper bound on s
There is no (123, 148, large)-net in base 9, because
- 23 times m-reduction [i] would yield (123, 125, large)-net in base 9, but