Best Known (67, 67+16, s)-Nets in Base 3
(67, 67+16, 821)-Net over F3 — Constructive and digital
Digital (67, 83, 821)-net over F3, using
- 31 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(382, 821, F3, 16, 16) (dual of [(821, 16), 13054, 17]-NRT-code), using
(67, 67+16, 2650)-Net over F3 — Digital
Digital (67, 83, 2650)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(383, 2650, F3, 2, 16) (dual of [(2650, 2), 5217, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(383, 3286, F3, 2, 16) (dual of [(3286, 2), 6489, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(383, 6572, F3, 16) (dual of [6572, 6489, 17]-code), using
- construction XX applied to Ce(15) ⊂ Ce(13) ⊂ Ce(12) [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(365, 6561, F3, 13) (dual of [6561, 6496, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(31, 10, F3, 1) (dual of [10, 9, 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
- linear OA(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(15) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- OOA 2-folding [i] based on linear OA(383, 6572, F3, 16) (dual of [6572, 6489, 17]-code), using
- discarding factors / shortening the dual code based on linear OOA(383, 3286, F3, 2, 16) (dual of [(3286, 2), 6489, 17]-NRT-code), using
(67, 67+16, 167792)-Net in Base 3 — Upper bound on s
There is no (67, 83, 167793)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 3990 850356 973552 708779 514587 510128 843169 > 383 [i]