Best Known (44, 44+10, s)-Nets in Base 9
(44, 44+10, 106298)-Net over F9 — Constructive and digital
Digital (44, 54, 106298)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 10)-net over F9, using
- net from sequence [i] based on digital (0, 9)-sequence over F9, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 0 and N(F) ≥ 10, using
- the rational function field F9(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 9)-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 (0, 5, 10)-net over F9, using
(44, 44+10, 531465)-Net over F9 — Digital
Digital (44, 54, 531465)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(954, 531465, F9, 10) (dual of [531465, 531411, 11]-code), using
- construction XX applied to Ce(9) ⊂ Ce(5) ⊂ 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(931, 531441, F9, 6) (dual of [531441, 531410, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(94, 23, F9, 3) (dual of [23, 19, 4]-code or 23-cap in PG(3,9)), using
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(9) ⊂ Ce(5) ⊂ Ce(4) [i] based on
(44, 44+10, large)-Net in Base 9 — Upper bound on s
There is no (44, 54, large)-net in base 9, because
- 8 times m-reduction [i] would yield (44, 46, large)-net in base 9, but