Best Known (41, 41+13, s)-Nets in Base 9
(41, 41+13, 2189)-Net over F9 — Constructive and digital
Digital (41, 54, 2189)-net over F9, using
- 91 times duplication [i] based on digital (40, 53, 2189)-net over F9, using
- net defined by OOA [i] based on linear OOA(953, 2189, F9, 13, 13) (dual of [(2189, 13), 28404, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(953, 13135, F9, 13) (dual of [13135, 13082, 14]-code), using
- 1 times code embedding in larger space [i] based on linear OA(952, 13134, F9, 13) (dual of [13134, 13082, 14]-code), using
- trace code [i] based on linear OA(8126, 6567, F81, 13) (dual of [6567, 6541, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(8125, 6562, F81, 13) (dual of [6562, 6537, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- 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]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- trace code [i] based on linear OA(8126, 6567, F81, 13) (dual of [6567, 6541, 14]-code), using
- 1 times code embedding in larger space [i] based on linear OA(952, 13134, F9, 13) (dual of [13134, 13082, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(953, 13135, F9, 13) (dual of [13135, 13082, 14]-code), using
- net defined by OOA [i] based on linear OOA(953, 2189, F9, 13, 13) (dual of [(2189, 13), 28404, 14]-NRT-code), using
(41, 41+13, 13138)-Net over F9 — Digital
Digital (41, 54, 13138)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(954, 13138, F9, 13) (dual of [13138, 13084, 14]-code), using
- trace code [i] based on linear OA(8127, 6569, F81, 13) (dual of [6569, 6542, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(8125, 6561, F81, 13) (dual of [6561, 6536, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(812, 8, F81, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,81)), using
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- Reed–Solomon code RS(79,81) [i]
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- trace code [i] based on linear OA(8127, 6569, F81, 13) (dual of [6569, 6542, 14]-code), using
(41, 41+13, large)-Net in Base 9 — Upper bound on s
There is no (41, 54, large)-net in base 9, because
- 11 times m-reduction [i] would yield (41, 43, large)-net in base 9, but