Best Known (86, 86+15, s)-Nets in Base 3
(86, 86+15, 8436)-Net over F3 — Constructive and digital
Digital (86, 101, 8436)-net over F3, using
- net defined by OOA [i] based on linear OOA(3101, 8436, F3, 15, 15) (dual of [(8436, 15), 126439, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3101, 59053, F3, 15) (dual of [59053, 58952, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3101, 59059, F3, 15) (dual of [59059, 58958, 16]-code), using
- 1 times truncation [i] based on linear OA(3102, 59060, F3, 16) (dual of [59060, 58958, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(3101, 59049, F3, 16) (dual of [59049, 58948, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(391, 59049, F3, 14) (dual of [59049, 58958, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(31, 11, F3, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 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(3102, 59060, F3, 16) (dual of [59060, 58958, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3101, 59059, F3, 15) (dual of [59059, 58958, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3101, 59053, F3, 15) (dual of [59053, 58952, 16]-code), using
(86, 86+15, 22823)-Net over F3 — Digital
Digital (86, 101, 22823)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3101, 22823, F3, 2, 15) (dual of [(22823, 2), 45545, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3101, 29529, F3, 2, 15) (dual of [(29529, 2), 58957, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3101, 59058, F3, 15) (dual of [59058, 58957, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3101, 59059, F3, 15) (dual of [59059, 58958, 16]-code), using
- 1 times truncation [i] based on linear OA(3102, 59060, F3, 16) (dual of [59060, 58958, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(3101, 59049, F3, 16) (dual of [59049, 58948, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(391, 59049, F3, 14) (dual of [59049, 58958, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(31, 11, F3, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 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(3102, 59060, F3, 16) (dual of [59060, 58958, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3101, 59059, F3, 15) (dual of [59059, 58958, 16]-code), using
- OOA 2-folding [i] based on linear OA(3101, 59058, F3, 15) (dual of [59058, 58957, 16]-code), using
- discarding factors / shortening the dual code based on linear OOA(3101, 29529, F3, 2, 15) (dual of [(29529, 2), 58957, 16]-NRT-code), using
(86, 86+15, large)-Net in Base 3 — Upper bound on s
There is no (86, 101, large)-net in base 3, because
- 13 times m-reduction [i] would yield (86, 88, large)-net in base 3, but