Best Known (34, 34+9, s)-Nets in Base 3
(34, 34+9, 548)-Net over F3 — Constructive and digital
Digital (34, 43, 548)-net over F3, using
- net defined by OOA [i] based on linear OOA(343, 548, F3, 9, 9) (dual of [(548, 9), 4889, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(343, 2193, F3, 9) (dual of [2193, 2150, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(343, 2194, F3, 9) (dual of [2194, 2151, 10]-code), using
- 1 times truncation [i] based on linear OA(344, 2195, F3, 10) (dual of [2195, 2151, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(343, 2187, F3, 10) (dual of [2187, 2144, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(336, 2187, F3, 8) (dual of [2187, 2151, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(31, 8, F3, 1) (dual of [8, 7, 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(9) ⊂ Ce(7) [i] based on
- 1 times truncation [i] based on linear OA(344, 2195, F3, 10) (dual of [2195, 2151, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(343, 2194, F3, 9) (dual of [2194, 2151, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(343, 2193, F3, 9) (dual of [2193, 2150, 10]-code), using
(34, 34+9, 1226)-Net over F3 — Digital
Digital (34, 43, 1226)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(343, 1226, F3, 9) (dual of [1226, 1183, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(343, 2188, F3, 9) (dual of [2188, 2145, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(343, 2188, F3, 9) (dual of [2188, 2145, 10]-code), using
(34, 34+9, 113183)-Net in Base 3 — Upper bound on s
There is no (34, 43, 113184)-net in base 3, because
- 1 times m-reduction [i] would yield (34, 42, 113184)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 109 421486 619356 305153 > 342 [i]