Best Known (85−15, 85, s)-Nets in Base 5
(85−15, 85, 11161)-Net over F5 — Constructive and digital
Digital (70, 85, 11161)-net over F5, using
- net defined by OOA [i] based on linear OOA(585, 11161, F5, 15, 15) (dual of [(11161, 15), 167330, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(585, 78128, F5, 15) (dual of [78128, 78043, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(585, 78132, F5, 15) (dual of [78132, 78047, 16]-code), using
- 1 times truncation [i] based on linear OA(586, 78133, F5, 16) (dual of [78133, 78047, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(585, 78125, F5, 16) (dual of [78125, 78040, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(578, 78125, F5, 14) (dual of [78125, 78047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(51, 8, F5, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- 1 times truncation [i] based on linear OA(586, 78133, F5, 16) (dual of [78133, 78047, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(585, 78132, F5, 15) (dual of [78132, 78047, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(585, 78128, F5, 15) (dual of [78128, 78043, 16]-code), using
(85−15, 85, 46525)-Net over F5 — Digital
Digital (70, 85, 46525)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(585, 46525, F5, 15) (dual of [46525, 46440, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(585, 78126, F5, 15) (dual of [78126, 78041, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(585, 78126, F5, 15) (dual of [78126, 78041, 16]-code), using
(85−15, 85, large)-Net in Base 5 — Upper bound on s
There is no (70, 85, large)-net in base 5, because
- 13 times m-reduction [i] would yield (70, 72, large)-net in base 5, but