Best Known (103, 103+18, s)-Nets in Base 3
(103, 103+18, 6562)-Net over F3 — Constructive and digital
Digital (103, 121, 6562)-net over F3, using
- net defined by OOA [i] based on linear OOA(3121, 6562, F3, 18, 18) (dual of [(6562, 18), 117995, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(3121, 59058, F3, 18) (dual of [59058, 58937, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3121, 59059, F3, 18) (dual of [59059, 58938, 19]-code), using
- 1 times truncation [i] based on linear OA(3122, 59060, F3, 19) (dual of [59060, 58938, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(3121, 59049, F3, 19) (dual of [59049, 58928, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3111, 59049, F3, 17) (dual of [59049, 58938, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(31, 11, F3, 1) (dual of [11, 10, 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(16) [i] based on
- 1 times truncation [i] based on linear OA(3122, 59060, F3, 19) (dual of [59060, 58938, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3121, 59059, F3, 18) (dual of [59059, 58938, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(3121, 59058, F3, 18) (dual of [59058, 58937, 19]-code), using
(103, 103+18, 19686)-Net over F3 — Digital
Digital (103, 121, 19686)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3121, 19686, F3, 3, 18) (dual of [(19686, 3), 58937, 19]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3121, 59058, F3, 18) (dual of [59058, 58937, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3121, 59059, F3, 18) (dual of [59059, 58938, 19]-code), using
- 1 times truncation [i] based on linear OA(3122, 59060, F3, 19) (dual of [59060, 58938, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(3121, 59049, F3, 19) (dual of [59049, 58928, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3111, 59049, F3, 17) (dual of [59049, 58938, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(31, 11, F3, 1) (dual of [11, 10, 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(16) [i] based on
- 1 times truncation [i] based on linear OA(3122, 59060, F3, 19) (dual of [59060, 58938, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3121, 59059, F3, 18) (dual of [59059, 58938, 19]-code), using
- OOA 3-folding [i] based on linear OA(3121, 59058, F3, 18) (dual of [59058, 58937, 19]-code), using
(103, 103+18, 5387049)-Net in Base 3 — Upper bound on s
There is no (103, 121, 5387050)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 5391 035028 354727 627795 545414 348598 054533 075713 948575 218821 > 3121 [i]