Best Known (42−11, 42, s)-Nets in Base 9
(42−11, 42, 2625)-Net over F9 — Constructive and digital
Digital (31, 42, 2625)-net over F9, using
- net defined by OOA [i] based on linear OOA(942, 2625, F9, 11, 11) (dual of [(2625, 11), 28833, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(942, 13126, F9, 11) (dual of [13126, 13084, 12]-code), using
- trace code [i] based on linear OA(8121, 6563, F81, 11) (dual of [6563, 6542, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(8121, 6561, F81, 11) (dual of [6561, 6540, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(8119, 6561, F81, 10) (dual of [6561, 6542, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(810, 2, F81, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- trace code [i] based on linear OA(8121, 6563, F81, 11) (dual of [6563, 6542, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(942, 13126, F9, 11) (dual of [13126, 13084, 12]-code), using
(42−11, 42, 11524)-Net over F9 — Digital
Digital (31, 42, 11524)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(942, 11524, F9, 11) (dual of [11524, 11482, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(942, 13124, F9, 11) (dual of [13124, 13082, 12]-code), using
- trace code [i] based on linear OA(8121, 6562, F81, 11) (dual of [6562, 6541, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- trace code [i] based on linear OA(8121, 6562, F81, 11) (dual of [6562, 6541, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(942, 13124, F9, 11) (dual of [13124, 13082, 12]-code), using
(42−11, 42, large)-Net in Base 9 — Upper bound on s
There is no (31, 42, large)-net in base 9, because
- 9 times m-reduction [i] would yield (31, 33, large)-net in base 9, but