Best Known (134−16, 134, s)-Nets in Base 3
(134−16, 134, 199292)-Net over F3 — Constructive and digital
Digital (118, 134, 199292)-net over F3, using
- 32 times duplication [i] based on digital (116, 132, 199292)-net over F3, using
- net defined by OOA [i] based on linear OOA(3132, 199292, F3, 16, 16) (dual of [(199292, 16), 3188540, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(3132, 1594336, F3, 16) (dual of [1594336, 1594204, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3132, 1594337, F3, 16) (dual of [1594337, 1594205, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(3131, 1594323, F3, 16) (dual of [1594323, 1594192, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(3118, 1594323, F3, 14) (dual of [1594323, 1594205, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(31, 14, F3, 1) (dual of [14, 13, 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(3132, 1594337, F3, 16) (dual of [1594337, 1594205, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(3132, 1594336, F3, 16) (dual of [1594336, 1594204, 17]-code), using
- net defined by OOA [i] based on linear OOA(3132, 199292, F3, 16, 16) (dual of [(199292, 16), 3188540, 17]-NRT-code), using
(134−16, 134, 427458)-Net over F3 — Digital
Digital (118, 134, 427458)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3134, 427458, F3, 3, 16) (dual of [(427458, 3), 1282240, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3134, 531446, F3, 3, 16) (dual of [(531446, 3), 1594204, 17]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3133, 531446, F3, 3, 16) (dual of [(531446, 3), 1594205, 17]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3133, 1594338, F3, 16) (dual of [1594338, 1594205, 17]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3132, 1594337, F3, 16) (dual of [1594337, 1594205, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(3131, 1594323, F3, 16) (dual of [1594323, 1594192, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(3118, 1594323, F3, 14) (dual of [1594323, 1594205, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(31, 14, F3, 1) (dual of [14, 13, 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 code embedding in larger space [i] based on linear OA(3132, 1594337, F3, 16) (dual of [1594337, 1594205, 17]-code), using
- OOA 3-folding [i] based on linear OA(3133, 1594338, F3, 16) (dual of [1594338, 1594205, 17]-code), using
- 31 times duplication [i] based on linear OOA(3133, 531446, F3, 3, 16) (dual of [(531446, 3), 1594205, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3134, 531446, F3, 3, 16) (dual of [(531446, 3), 1594204, 17]-NRT-code), using
(134−16, 134, large)-Net in Base 3 — Upper bound on s
There is no (118, 134, large)-net in base 3, because
- 14 times m-reduction [i] would yield (118, 120, large)-net in base 3, but