Best Known (112−11, 112, s)-Nets in Base 9
(112−11, 112, 5746932)-Net over F9 — Constructive and digital
Digital (101, 112, 5746932)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (25, 30, 2391492)-net over F9, using
- net defined by OOA [i] based on linear OOA(930, 2391492, F9, 5, 5) (dual of [(2391492, 5), 11957430, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(930, 4782985, F9, 5) (dual of [4782985, 4782955, 6]-code), using
- construction X applied to C([0,2]) ⊂ C([0,1]) [i] based on
- linear OA(929, 4782970, F9, 5) (dual of [4782970, 4782941, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 914−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(915, 4782970, F9, 3) (dual of [4782970, 4782955, 4]-code or 4782970-cap in PG(14,9)), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 914−1, defining interval I = [0,1], and minimum distance d ≥ |{−1,0,1}|+1 = 4 (BCH-bound) [i]
- linear OA(91, 15, F9, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,2]) ⊂ C([0,1]) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(930, 4782985, F9, 5) (dual of [4782985, 4782955, 6]-code), using
- net defined by OOA [i] based on linear OOA(930, 2391492, F9, 5, 5) (dual of [(2391492, 5), 11957430, 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 (25, 30, 2391492)-net over F9, using
(112−11, 112, large)-Net over F9 — Digital
Digital (101, 112, large)-net over F9, using
- 6 times m-reduction [i] based on digital (101, 118, large)-net over F9, using
(112−11, 112, large)-Net in Base 9 — Upper bound on s
There is no (101, 112, large)-net in base 9, because
- 9 times m-reduction [i] would yield (101, 103, large)-net in base 9, but