Best Known (111, 111+11, s)-Nets in Base 9
(111, 111+11, 7158757)-Net over F9 — Constructive and digital
Digital (111, 122, 7158757)-net over F9, using
- generalized (u, u+v)-construction [i] 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
- 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
- digital (71, 82, 3355440)-net over F9, using
- trace code for nets [i] based on digital (30, 41, 1677720)-net over F81, using
- net defined by OOA [i] based on linear OOA(8141, 1677720, F81, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(8141, 8388601, F81, 11) (dual of [8388601, 8388560, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(8141, large, F81, 11) (dual of [large, large−41, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8141, large, F81, 11) (dual of [large, large−41, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(8141, 8388601, F81, 11) (dual of [8388601, 8388560, 12]-code), using
- net defined by OOA [i] based on linear OOA(8141, 1677720, F81, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- trace code for nets [i] based on digital (30, 41, 1677720)-net over F81, using
- digital (8, 11, 1411830)-net over F9, using
(111, 111+11, large)-Net over F9 — Digital
Digital (111, 122, large)-net over F9, using
- t-expansion [i] based on digital (110, 122, large)-net over F9, using
- 7 times m-reduction [i] based on digital (110, 129, large)-net over F9, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(9129, large, F9, 19) (dual of [large, large−129, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 916−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(9129, large, F9, 19) (dual of [large, large−129, 20]-code), using
- 7 times m-reduction [i] based on digital (110, 129, large)-net over F9, using
(111, 111+11, large)-Net in Base 9 — Upper bound on s
There is no (111, 122, large)-net in base 9, because
- 9 times m-reduction [i] would yield (111, 113, large)-net in base 9, but