Best Known (123, 123+16, s)-Nets in Base 3
(123, 123+16, 199295)-Net over F3 — Constructive and digital
Digital (123, 139, 199295)-net over F3, using
- net defined by OOA [i] based on linear OOA(3139, 199295, F3, 16, 16) (dual of [(199295, 16), 3188581, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(3139, 1594360, F3, 16) (dual of [1594360, 1594221, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3139, 1594364, F3, 16) (dual of [1594364, 1594225, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [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(392, 1594323, F3, 11) (dual of [1594323, 1594231, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(3139, 1594364, F3, 16) (dual of [1594364, 1594225, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(3139, 1594360, F3, 16) (dual of [1594360, 1594221, 17]-code), using
(123, 123+16, 531454)-Net over F3 — Digital
Digital (123, 139, 531454)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3139, 531454, F3, 3, 16) (dual of [(531454, 3), 1594223, 17]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3139, 1594362, F3, 16) (dual of [1594362, 1594223, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(3139, 1594364, F3, 16) (dual of [1594364, 1594225, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [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(392, 1594323, F3, 11) (dual of [1594323, 1594231, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(3139, 1594364, F3, 16) (dual of [1594364, 1594225, 17]-code), using
- OOA 3-folding [i] based on linear OA(3139, 1594362, F3, 16) (dual of [1594362, 1594223, 17]-code), using
(123, 123+16, large)-Net in Base 3 — Upper bound on s
There is no (123, 139, large)-net in base 3, because
- 14 times m-reduction [i] would yield (123, 125, large)-net in base 3, but