Best Known (122−31, 122, s)-Nets in Base 3
(122−31, 122, 264)-Net over F3 — Constructive and digital
Digital (91, 122, 264)-net over F3, using
- 1 times m-reduction [i] based on digital (91, 123, 264)-net over F3, using
- trace code for nets [i] based on digital (9, 41, 88)-net over F27, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 9 and N(F) ≥ 88, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- trace code for nets [i] based on digital (9, 41, 88)-net over F27, using
(122−31, 122, 546)-Net over F3 — Digital
Digital (91, 122, 546)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3122, 546, F3, 31) (dual of [546, 424, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3122, 742, F3, 31) (dual of [742, 620, 32]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3121, 741, F3, 31) (dual of [741, 620, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- linear OA(3118, 729, F3, 31) (dual of [729, 611, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3109, 729, F3, 28) (dual of [729, 620, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(33, 12, F3, 2) (dual of [12, 9, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3121, 741, F3, 31) (dual of [741, 620, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3122, 742, F3, 31) (dual of [742, 620, 32]-code), using
(122−31, 122, 22658)-Net in Base 3 — Upper bound on s
There is no (91, 122, 22659)-net in base 3, because
- 1 times m-reduction [i] would yield (91, 121, 22659)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 5391 724425 999536 580376 714075 418853 950103 475822 265110 153971 > 3121 [i]