Best Known (39, 39+13, s)-Nets in Base 9
(39, 39+13, 2188)-Net over F9 — Constructive and digital
Digital (39, 52, 2188)-net over F9, using
- net defined by OOA [i] based on linear OOA(952, 2188, F9, 13, 13) (dual of [(2188, 13), 28392, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(952, 13129, F9, 13) (dual of [13129, 13077, 14]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code 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(952, 13129, F9, 13) (dual of [13129, 13077, 14]-code), using
(39, 39+13, 13134)-Net over F9 — Digital
Digital (39, 52, 13134)-net over F9, using
- embedding of OOA with Gilbert–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
(39, 39+13, large)-Net in Base 9 — Upper bound on s
There is no (39, 52, large)-net in base 9, because
- 11 times m-reduction [i] would yield (39, 41, large)-net in base 9, but