Best Known (112, 112+3, s)-Nets in Base 9
(112, 112+3, large)-Net over F9 — Constructive and digital
Digital (112, 115, large)-net over F9, using
- t-expansion [i] based on digital (107, 115, large)-net over F9, using
- 2 times m-reduction [i] based on digital (107, 117, large)-net over F9, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 1, 956593)-net over F9, using
- s-reduction based on digital (0, 1, s)-net over F9 with arbitrarily large s, using
- digital (0, 1, 956593)-net over F9 (see above)
- digital (0, 1, 956593)-net over F9 (see above)
- digital (0, 1, 956593)-net over F9 (see above)
- digital (6, 8, 956593)-net over F9, using
- s-reduction based on digital (6, 8, 5380840)-net over F9, using
- digital (6, 8, 956593)-net over F9 (see above)
- digital (8, 11, 956593)-net over F9, using
- s-reduction based on digital (8, 11, 1411830)-net over F9, using
- net defined by OOA [i] based on linear OOA(911, 1411830, F9, 3, 3) (dual of [(1411830, 3), 4235479, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(911, 1411830, F9, 2, 3) (dual of [(1411830, 2), 2823649, 4]-NRT-code), using
- OAs with strength 3, b ≠ 2, and m > 3 are always embeddable [i] based on linear OA(911, 1411830, F9, 3) (dual of [1411830, 1411819, 4]-code or 1411830-cap in PG(10,9)), using
- appending kth column [i] based on linear OOA(911, 1411830, F9, 2, 3) (dual of [(1411830, 2), 2823649, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(911, 1411830, F9, 3, 3) (dual of [(1411830, 3), 4235479, 4]-NRT-code), using
- s-reduction based on digital (8, 11, 1411830)-net over F9, using
- digital (24, 29, 956593)-net over F9, using
- s-reduction based on digital (24, 29, 2391487)-net over F9, using
- net defined by OOA [i] based on linear OOA(929, 2391487, F9, 5, 5) (dual of [(2391487, 5), 11957406, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(929, 4782975, F9, 5) (dual of [4782975, 4782946, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(929, 4782976, F9, 5) (dual of [4782976, 4782947, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(929, 4782969, F9, 5) (dual of [4782969, 4782940, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(922, 4782969, F9, 4) (dual of [4782969, 4782947, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(90, 7, F9, 0) (dual of [7, 7, 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
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(929, 4782976, F9, 5) (dual of [4782976, 4782947, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(929, 4782975, F9, 5) (dual of [4782975, 4782946, 6]-code), using
- net defined by OOA [i] based on linear OOA(929, 2391487, F9, 5, 5) (dual of [(2391487, 5), 11957406, 6]-NRT-code), using
- s-reduction based on digital (24, 29, 2391487)-net over F9, using
- digital (47, 57, 956593)-net over F9, using
- net defined by OOA [i] based on linear OOA(957, 956593, F9, 10, 10) (dual of [(956593, 10), 9565873, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(957, 4782965, F9, 10) (dual of [4782965, 4782908, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(957, 4782969, F9, 10) (dual of [4782969, 4782912, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(957, 4782969, F9, 10) (dual of [4782969, 4782912, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(957, 4782965, F9, 10) (dual of [4782965, 4782908, 11]-code), using
- net defined by OOA [i] based on linear OOA(957, 956593, F9, 10, 10) (dual of [(956593, 10), 9565873, 11]-NRT-code), using
- digital (0, 1, 956593)-net over F9, using
- generalized (u, u+v)-construction [i] based on
- 2 times m-reduction [i] based on digital (107, 117, large)-net over F9, using
(112, 112+3, large)-Net in Base 9 — Upper bound on s
There is no (112, 115, large)-net in base 9, because
- 1 times m-reduction [i] would yield (112, 114, large)-net in base 9, but