Best Known (140, 140+18, s)-Nets in Base 3
(140, 140+18, 177150)-Net over F3 — Constructive and digital
Digital (140, 158, 177150)-net over F3, using
- net defined by OOA [i] based on linear OOA(3158, 177150, F3, 18, 18) (dual of [(177150, 18), 3188542, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(3158, 1594350, F3, 18) (dual of [1594350, 1594192, 19]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(3157, 1594323, F3, 19) (dual of [1594323, 1594166, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 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(31, 27, F3, 1) (dual of [27, 26, 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(18) ⊂ Ce(15) [i] based on
- OA 9-folding and stacking [i] based on linear OA(3158, 1594350, F3, 18) (dual of [1594350, 1594192, 19]-code), using
(140, 140+18, 531450)-Net over F3 — Digital
Digital (140, 158, 531450)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3158, 531450, F3, 3, 18) (dual of [(531450, 3), 1594192, 19]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3158, 1594350, F3, 18) (dual of [1594350, 1594192, 19]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(3157, 1594323, F3, 19) (dual of [1594323, 1594166, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 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(31, 27, F3, 1) (dual of [27, 26, 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(18) ⊂ Ce(15) [i] based on
- OOA 3-folding [i] based on linear OA(3158, 1594350, F3, 18) (dual of [1594350, 1594192, 19]-code), using
(140, 140+18, large)-Net in Base 3 — Upper bound on s
There is no (140, 158, large)-net in base 3, because
- 16 times m-reduction [i] would yield (140, 142, large)-net in base 3, but