Best Known (25, 32, s)-Nets in Base 9
(25, 32, 19687)-Net over F9 — Constructive and digital
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
(25, 32, 59062)-Net over F9 — Digital
Digital (25, 32, 59062)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [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
(25, 32, large)-Net in Base 9 — Upper bound on s
There is no (25, 32, large)-net in base 9, because
- 5 times m-reduction [i] would yield (25, 27, large)-net in base 9, but