Best Known (64, 64+13, s)-Nets in Base 9
(64, 64+13, 177149)-Net over F9 — Constructive and digital
Digital (64, 77, 177149)-net over F9, using
- 91 times duplication [i] based on digital (63, 76, 177149)-net over F9, using
- net defined by OOA [i] based on linear OOA(976, 177149, F9, 13, 13) (dual of [(177149, 13), 2302861, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(976, 1062895, F9, 13) (dual of [1062895, 1062819, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(976, 1062898, F9, 13) (dual of [1062898, 1062822, 14]-code), using
- trace code [i] based on linear OA(8138, 531449, F81, 13) (dual of [531449, 531411, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(8137, 531442, F81, 13) (dual of [531442, 531405, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(8131, 531442, F81, 11) (dual of [531442, 531411, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(811, 7, F81, 1) (dual of [7, 6, 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(8138, 531449, F81, 13) (dual of [531449, 531411, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(976, 1062898, F9, 13) (dual of [1062898, 1062822, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(976, 1062895, F9, 13) (dual of [1062895, 1062819, 14]-code), using
- net defined by OOA [i] based on linear OOA(976, 177149, F9, 13, 13) (dual of [(177149, 13), 2302861, 14]-NRT-code), using
(64, 64+13, 1062900)-Net over F9 — Digital
Digital (64, 77, 1062900)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(977, 1062900, F9, 13) (dual of [1062900, 1062823, 14]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(976, 1062898, F9, 13) (dual of [1062898, 1062822, 14]-code), using
- trace code [i] based on linear OA(8138, 531449, F81, 13) (dual of [531449, 531411, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(8137, 531442, F81, 13) (dual of [531442, 531405, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(8131, 531442, F81, 11) (dual of [531442, 531411, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(811, 7, F81, 1) (dual of [7, 6, 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(8138, 531449, F81, 13) (dual of [531449, 531411, 14]-code), using
- linear OA(976, 1062899, F9, 12) (dual of [1062899, 1062823, 13]-code), using Gilbert–Varšamov bound and bm = 976 > Vbs−1(k−1) = 420 939534 469201 543845 633548 125825 226705 965928 413623 471188 022427 072977 [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(976, 1062898, F9, 13) (dual of [1062898, 1062822, 14]-code), using
- construction X with Varšamov bound [i] based on
(64, 64+13, large)-Net in Base 9 — Upper bound on s
There is no (64, 77, large)-net in base 9, because
- 11 times m-reduction [i] would yield (64, 66, large)-net in base 9, but