Best Known (122, 122+25, s)-Nets in Base 9
(122, 122+25, 88574)-Net over F9 — Constructive and digital
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
(122, 122+25, 1062890)-Net over F9 — Digital
Digital (122, 147, 1062890)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9147, 1062890, F9, 25) (dual of [1062890, 1062743, 26]-code), using
- construction X with Varšamov bound [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
- linear OA(9146, 1062889, F9, 24) (dual of [1062889, 1062743, 25]-code), using Gilbert–Varšamov bound and bm = 9146 > Vbs−1(k−1) = 92829 425049 711883 295251 287327 941465 649616 246641 454212 255187 342372 204832 645110 698004 745170 062357 651290 178691 853317 006369 236622 023741 643329 [i]
- linear OA(90, 1, F9, 0) (dual of [1, 1, 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(9146, 1062888, F9, 25) (dual of [1062888, 1062742, 26]-code), using
- construction X with Varšamov bound [i] based on
(122, 122+25, large)-Net in Base 9 — Upper bound on s
There is no (122, 147, large)-net in base 9, because
- 23 times m-reduction [i] would yield (122, 124, large)-net in base 9, but