Best Known (78−19, 78, s)-Nets in Base 9
(78−19, 78, 1459)-Net over F9 — Constructive and digital
Digital (59, 78, 1459)-net over F9, using
- 92 times duplication [i] based on digital (57, 76, 1459)-net over F9, using
- net defined by OOA [i] based on linear OOA(976, 1459, F9, 19, 19) (dual of [(1459, 19), 27645, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(976, 13132, F9, 19) (dual of [13132, 13056, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(976, 13134, F9, 19) (dual of [13134, 13058, 20]-code), using
- trace code [i] based on linear OA(8138, 6567, F81, 19) (dual of [6567, 6529, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(8137, 6562, F81, 19) (dual of [6562, 6525, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(8133, 6562, F81, 17) (dual of [6562, 6529, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (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([0,9]) ⊂ C([0,8]) [i] based on
- trace code [i] based on linear OA(8138, 6567, F81, 19) (dual of [6567, 6529, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(976, 13134, F9, 19) (dual of [13134, 13058, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(976, 13132, F9, 19) (dual of [13132, 13056, 20]-code), using
- net defined by OOA [i] based on linear OOA(976, 1459, F9, 19, 19) (dual of [(1459, 19), 27645, 20]-NRT-code), using
(78−19, 78, 13138)-Net over F9 — Digital
Digital (59, 78, 13138)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(978, 13138, F9, 19) (dual of [13138, 13060, 20]-code), using
- trace code [i] based on linear OA(8139, 6569, F81, 19) (dual of [6569, 6530, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(8137, 6561, F81, 19) (dual of [6561, 6524, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(8131, 6561, F81, 16) (dual of [6561, 6530, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(18) ⊂ Ce(15) [i] based on
- trace code [i] based on linear OA(8139, 6569, F81, 19) (dual of [6569, 6530, 20]-code), using
(78−19, 78, large)-Net in Base 9 — Upper bound on s
There is no (59, 78, large)-net in base 9, because
- 17 times m-reduction [i] would yield (59, 61, large)-net in base 9, but