Best Known (122−19, 122, s)-Nets in Base 3
(122−19, 122, 6562)-Net over F3 — Constructive and digital
Digital (103, 122, 6562)-net over F3, using
- net defined by OOA [i] based on linear OOA(3122, 6562, F3, 19, 19) (dual of [(6562, 19), 124556, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3122, 59059, F3, 19) (dual of [59059, 58937, 20]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(3122, 59060, F3, 19) (dual of [59060, 58938, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3122, 59059, F3, 19) (dual of [59059, 58937, 20]-code), using
(122−19, 122, 19570)-Net over F3 — Digital
Digital (103, 122, 19570)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3122, 19570, F3, 3, 19) (dual of [(19570, 3), 58588, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3122, 19686, F3, 3, 19) (dual of [(19686, 3), 58936, 20]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3122, 59058, F3, 19) (dual of [59058, 58936, 20]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(3122, 59060, F3, 19) (dual of [59060, 58938, 20]-code), using
- OOA 3-folding [i] based on linear OA(3122, 59058, F3, 19) (dual of [59058, 58936, 20]-code), using
- discarding factors / shortening the dual code based on linear OOA(3122, 19686, F3, 3, 19) (dual of [(19686, 3), 58936, 20]-NRT-code), using
(122−19, 122, 5387049)-Net in Base 3 — Upper bound on s
There is no (103, 122, 5387050)-net in base 3, because
- 1 times m-reduction [i] would yield (103, 121, 5387050)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 5391 035028 354727 627795 545414 348598 054533 075713 948575 218821 > 3121 [i]