Best Known (69, 91, s)-Nets in Base 3
(69, 91, 328)-Net over F3 — Constructive and digital
Digital (69, 91, 328)-net over F3, using
- 1 times m-reduction [i] based on digital (69, 92, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 23, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- trace code for nets [i] based on digital (0, 23, 82)-net over F81, using
(69, 91, 565)-Net over F3 — Digital
Digital (69, 91, 565)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(391, 565, F3, 22) (dual of [565, 474, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(391, 748, F3, 22) (dual of [748, 657, 23]-code), using
- construction XX applied to Ce(21) ⊂ Ce(18) ⊂ Ce(16) [i] based on
- linear OA(385, 729, F3, 22) (dual of [729, 644, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(373, 729, F3, 19) (dual of [729, 656, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(367, 729, F3, 17) (dual of [729, 662, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(34, 17, F3, 2) (dual of [17, 13, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- linear OA(31, 2, F3, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(21) ⊂ Ce(18) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(391, 748, F3, 22) (dual of [748, 657, 23]-code), using
(69, 91, 21720)-Net in Base 3 — Upper bound on s
There is no (69, 91, 21721)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 26 191508 521254 139571 524892 224058 965716 220443 > 391 [i]