Best Known (133−28, 133, s)-Nets in Base 3
(133−28, 133, 640)-Net over F3 — Constructive and digital
Digital (105, 133, 640)-net over F3, using
- 31 times duplication [i] based on digital (104, 132, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 33, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 33, 160)-net over F81, using
(133−28, 133, 1371)-Net over F3 — Digital
Digital (105, 133, 1371)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3133, 1371, F3, 28) (dual of [1371, 1238, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3133, 2208, F3, 28) (dual of [2208, 2075, 29]-code), using
- construction XX applied to Ce(27) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- linear OA(3127, 2187, F3, 28) (dual of [2187, 2060, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3113, 2187, F3, 25) (dual of [2187, 2074, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3106, 2187, F3, 23) (dual of [2187, 2081, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(34, 19, F3, 2) (dual of [19, 15, 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(27) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(3133, 2208, F3, 28) (dual of [2208, 2075, 29]-code), using
(133−28, 133, 103044)-Net in Base 3 — Upper bound on s
There is no (105, 133, 103045)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 2865 398193 090149 508235 533485 163837 695387 733879 774968 093738 078817 > 3133 [i]