Best Known (131, 131+15, s)-Nets in Base 9
(131, 131+15, 2416429)-Net over F9 — Constructive and digital
Digital (131, 146, 2416429)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (25, 32, 19687)-net over F9, using
- net defined by OOA [i] based on linear OOA(932, 19687, F9, 7, 7) (dual of [(19687, 7), 137777, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(932, 59062, F9, 7) (dual of [59062, 59030, 8]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(931, 59050, F9, 7) (dual of [59050, 59019, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 910−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(921, 59050, F9, 5) (dual of [59050, 59029, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 910−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(911, 12, F9, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,9)), using
- dual of repetition code with length 12 [i]
- linear OA(91, 12, F9, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(932, 59062, F9, 7) (dual of [59062, 59030, 8]-code), using
- net defined by OOA [i] based on linear OOA(932, 19687, F9, 7, 7) (dual of [(19687, 7), 137777, 8]-NRT-code), using
- digital (99, 114, 2396742)-net over F9, using
- trace code for nets [i] based on digital (42, 57, 1198371)-net over F81, using
- net defined by OOA [i] based on linear OOA(8157, 1198371, F81, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(8157, 8388598, F81, 15) (dual of [8388598, 8388541, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(8157, large, F81, 15) (dual of [large, large−57, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8157, large, F81, 15) (dual of [large, large−57, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(8157, 8388598, F81, 15) (dual of [8388598, 8388541, 16]-code), using
- net defined by OOA [i] based on linear OOA(8157, 1198371, F81, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- trace code for nets [i] based on digital (42, 57, 1198371)-net over F81, using
- digital (25, 32, 19687)-net over F9, using
(131, 131+15, large)-Net over F9 — Digital
Digital (131, 146, large)-net over F9, using
- 91 times duplication [i] based on digital (130, 145, large)-net over F9, using
- t-expansion [i] based on digital (124, 145, large)-net over F9, using
(131, 131+15, large)-Net in Base 9 — Upper bound on s
There is no (131, 146, large)-net in base 9, because
- 13 times m-reduction [i] would yield (131, 133, large)-net in base 9, but