Best Known (32−7, 32, s)-Nets in Base 3
(32−7, 32, 733)-Net over F3 — Constructive and digital
Digital (25, 32, 733)-net over F3, using
- net defined by OOA [i] based on linear OOA(332, 733, F3, 7, 7) (dual of [(733, 7), 5099, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(332, 2200, F3, 7) (dual of [2200, 2168, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- linear OA(329, 2187, F3, 7) (dual of [2187, 2158, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(315, 2187, F3, 4) (dual of [2187, 2172, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(332, 2200, F3, 7) (dual of [2200, 2168, 8]-code), using
(32−7, 32, 1179)-Net over F3 — Digital
Digital (25, 32, 1179)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(332, 1179, F3, 7) (dual of [1179, 1147, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(332, 2192, F3, 7) (dual of [2192, 2160, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([1,3]) [i] based on
- linear OA(329, 2188, F3, 7) (dual of [2188, 2159, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(328, 2188, F3, 3) (dual of [2188, 2160, 4]-code or 2188-cap in PG(27,3)), using the narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(33, 4, F3, 3) (dual of [4, 1, 4]-code or 4-arc in PG(2,3) or 4-cap in PG(2,3)), using
- dual of repetition code with length 4 [i]
- oval in PG(2, 3) [i]
- construction X applied to C([0,3]) ⊂ C([1,3]) [i] based on
- discarding factors / shortening the dual code based on linear OA(332, 2192, F3, 7) (dual of [2192, 2160, 8]-code), using
(32−7, 32, 77373)-Net in Base 3 — Upper bound on s
There is no (25, 32, 77374)-net in base 3, because
- 1 times m-reduction [i] would yield (25, 31, 77374)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 617 683470 719745 > 331 [i]