Best Known (82−16, 82, s)-Nets in Base 3
(82−16, 82, 821)-Net over F3 — Constructive and digital
Digital (66, 82, 821)-net over F3, using
- net defined by OOA [i] based on linear OOA(382, 821, F3, 16, 16) (dual of [(821, 16), 13054, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(382, 6568, F3, 16) (dual of [6568, 6486, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(382, 6570, F3, 16) (dual of [6570, 6488, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(381, 6561, F3, 16) (dual of [6561, 6480, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(373, 6561, F3, 14) (dual of [6561, 6488, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(31, 9, F3, 1) (dual of [9, 8, 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
- discarding factors / shortening the dual code based on linear OA(382, 6570, F3, 16) (dual of [6570, 6488, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(382, 6568, F3, 16) (dual of [6568, 6486, 17]-code), using
(82−16, 82, 2434)-Net over F3 — Digital
Digital (66, 82, 2434)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(382, 2434, F3, 2, 16) (dual of [(2434, 2), 4786, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(382, 3285, F3, 2, 16) (dual of [(3285, 2), 6488, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(382, 6570, F3, 16) (dual of [6570, 6488, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(381, 6561, F3, 16) (dual of [6561, 6480, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(373, 6561, F3, 14) (dual of [6561, 6488, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(31, 9, F3, 1) (dual of [9, 8, 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
- OOA 2-folding [i] based on linear OA(382, 6570, F3, 16) (dual of [6570, 6488, 17]-code), using
- discarding factors / shortening the dual code based on linear OOA(382, 3285, F3, 2, 16) (dual of [(3285, 2), 6488, 17]-NRT-code), using
(82−16, 82, 146261)-Net in Base 3 — Upper bound on s
There is no (66, 82, 146262)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 1330 300188 556450 126574 559187 422451 268817 > 382 [i]