Best Known (61−15, 61, s)-Nets in Base 9
(61−15, 61, 1876)-Net over F9 — Constructive and digital
Digital (46, 61, 1876)-net over F9, using
- 91 times duplication [i] based on digital (45, 60, 1876)-net over F9, using
- net defined by OOA [i] based on linear OOA(960, 1876, F9, 15, 15) (dual of [(1876, 15), 28080, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(960, 13133, F9, 15) (dual of [13133, 13073, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(960, 13134, F9, 15) (dual of [13134, 13074, 16]-code), using
- trace code [i] based on linear OA(8130, 6567, F81, 15) (dual of [6567, 6537, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(8129, 6562, F81, 15) (dual of [6562, 6533, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- 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(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([0,7]) ⊂ C([0,6]) [i] based on
- trace code [i] based on linear OA(8130, 6567, F81, 15) (dual of [6567, 6537, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(960, 13134, F9, 15) (dual of [13134, 13074, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(960, 13133, F9, 15) (dual of [13133, 13073, 16]-code), using
- net defined by OOA [i] based on linear OOA(960, 1876, F9, 15, 15) (dual of [(1876, 15), 28080, 16]-NRT-code), using
(61−15, 61, 13136)-Net over F9 — Digital
Digital (46, 61, 13136)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(961, 13136, F9, 15) (dual of [13136, 13075, 16]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(960, 13134, F9, 15) (dual of [13134, 13074, 16]-code), using
- trace code [i] based on linear OA(8130, 6567, F81, 15) (dual of [6567, 6537, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(8129, 6562, F81, 15) (dual of [6562, 6533, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- 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(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([0,7]) ⊂ C([0,6]) [i] based on
- trace code [i] based on linear OA(8130, 6567, F81, 15) (dual of [6567, 6537, 16]-code), using
- linear OA(960, 13135, F9, 14) (dual of [13135, 13075, 15]-code), using Gilbert–Varšamov bound and bm = 960 > Vbs−1(k−1) = 30 375571 288920 254702 038083 474426 714481 454719 794757 341489 [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(960, 13134, F9, 15) (dual of [13134, 13074, 16]-code), using
- construction X with Varšamov bound [i] based on
(61−15, 61, large)-Net in Base 9 — Upper bound on s
There is no (46, 61, large)-net in base 9, because
- 13 times m-reduction [i] would yield (46, 48, large)-net in base 9, but