Best Known (134−24, 134, s)-Nets in Base 3
(134−24, 134, 688)-Net over F3 — Constructive and digital
Digital (110, 134, 688)-net over F3, using
- t-expansion [i] based on digital (109, 134, 688)-net over F3, using
- 2 times m-reduction [i] based on digital (109, 136, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 34, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 34, 172)-net over F81, using
- 2 times m-reduction [i] based on digital (109, 136, 688)-net over F3, using
(134−24, 134, 3449)-Net over F3 — Digital
Digital (110, 134, 3449)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3134, 3449, F3, 24) (dual of [3449, 3315, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3134, 6585, F3, 24) (dual of [6585, 6451, 25]-code), using
- construction XX applied to Ce(24) ⊂ Ce(21) ⊂ Ce(19) [i] based on
- linear OA(3129, 6561, F3, 25) (dual of [6561, 6432, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3113, 6561, F3, 22) (dual of [6561, 6448, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3105, 6561, F3, 20) (dual of [6561, 6456, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(31, 20, F3, 1) (dual of [20, 19, 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
- linear OA(31, 4, F3, 1) (dual of [4, 3, 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 (see above)
- construction XX applied to Ce(24) ⊂ Ce(21) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3134, 6585, F3, 24) (dual of [6585, 6451, 25]-code), using
(134−24, 134, 562569)-Net in Base 3 — Upper bound on s
There is no (110, 134, 562570)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 8595 048724 726640 360562 017485 884724 175626 614710 622164 336973 409657 > 3134 [i]