Best Known (123−19, 123, s)-Nets in Base 3
(123−19, 123, 6562)-Net over F3 — Constructive and digital
Digital (104, 123, 6562)-net over F3, using
- 31 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(3122, 6562, F3, 19, 19) (dual of [(6562, 19), 124556, 20]-NRT-code), using
(123−19, 123, 19687)-Net over F3 — Digital
Digital (104, 123, 19687)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3123, 19687, F3, 3, 19) (dual of [(19687, 3), 58938, 20]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3123, 59061, F3, 19) (dual of [59061, 58938, 20]-code), using
- 1 times code embedding in larger space [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 code embedding in larger space [i] based on linear OA(3122, 59060, F3, 19) (dual of [59060, 58938, 20]-code), using
- OOA 3-folding [i] based on linear OA(3123, 59061, F3, 19) (dual of [59061, 58938, 20]-code), using
(123−19, 123, 6086456)-Net in Base 3 — Upper bound on s
There is no (104, 123, 6086457)-net in base 3, because
- 1 times m-reduction [i] would yield (104, 122, 6086457)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 16173 105140 263631 839660 960294 088139 004754 136577 365991 111635 > 3122 [i]