Best Known (55−10, 55, s)-Nets in Base 9
(55−10, 55, 106304)-Net over F9 — Constructive and digital
Digital (45, 55, 106304)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (1, 6, 16)-net over F9, using
- net from sequence [i] based on digital (1, 15)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 1 and N(F) ≥ 16, using
- net from sequence [i] based on digital (1, 15)-sequence over F9, using
- digital (39, 49, 106288)-net over F9, using
- net defined by OOA [i] based on linear OOA(949, 106288, F9, 10, 10) (dual of [(106288, 10), 1062831, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(949, 531440, F9, 10) (dual of [531440, 531391, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(949, 531441, F9, 10) (dual of [531441, 531392, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(949, 531441, F9, 10) (dual of [531441, 531392, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(949, 531440, F9, 10) (dual of [531440, 531391, 11]-code), using
- net defined by OOA [i] based on linear OOA(949, 106288, F9, 10, 10) (dual of [(106288, 10), 1062831, 11]-NRT-code), using
- digital (1, 6, 16)-net over F9, using
(55−10, 55, 531471)-Net over F9 — Digital
Digital (45, 55, 531471)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(955, 531471, F9, 10) (dual of [531471, 531416, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(4) [i] based on
- linear OA(949, 531441, F9, 10) (dual of [531441, 531392, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(925, 531441, F9, 5) (dual of [531441, 531416, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(96, 30, F9, 4) (dual of [30, 24, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(96, 72, F9, 4) (dual of [72, 66, 5]-code), using
- 1 times truncation [i] based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(96, 72, F9, 4) (dual of [72, 66, 5]-code), using
- construction X applied to Ce(9) ⊂ Ce(4) [i] based on
(55−10, 55, large)-Net in Base 9 — Upper bound on s
There is no (45, 55, large)-net in base 9, because
- 8 times m-reduction [i] would yield (45, 47, large)-net in base 9, but