Best Known (40, 53, s)-Nets in Base 9
(40, 53, 2189)-Net over F9 — Constructive and digital
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
(40, 53, 13136)-Net over F9 — Digital
Digital (40, 53, 13136)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(953, 13136, F9, 13) (dual of [13136, 13083, 14]-code), using
- construction X with Varšamov bound [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
- linear OA(952, 13135, F9, 12) (dual of [13135, 13083, 13]-code), using Gilbert–Varšamov bound and bm = 952 > Vbs−1(k−1) = 429959 584804 971312 035988 248374 137515 635195 613489 [i]
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(952, 13134, F9, 13) (dual of [13134, 13082, 14]-code), using
- construction X with Varšamov bound [i] based on
(40, 53, large)-Net in Base 9 — Upper bound on s
There is no (40, 53, large)-net in base 9, because
- 11 times m-reduction [i] would yield (40, 42, large)-net in base 9, but