Best Known (32, 32+5, s)-Nets in Base 3
(32, 32+5, 265726)-Net over F3 — Constructive and digital
Digital (32, 37, 265726)-net over F3, using
- net defined by OOA [i] based on linear OOA(337, 265726, F3, 5, 5) (dual of [(265726, 5), 1328593, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(337, 531453, F3, 5) (dual of [531453, 531416, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(337, 531441, F3, 5) (dual of [531441, 531404, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(325, 531441, F3, 4) (dual of [531441, 531416, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(30, 12, F3, 0) (dual of [12, 12, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(337, 531453, F3, 5) (dual of [531453, 531416, 6]-code), using
(32, 32+5, 482844)-Net over F3 — Digital
Digital (32, 37, 482844)-net over F3, using
- net defined by OOA [i] based on linear OOA(337, 482844, F3, 5, 5) (dual of [(482844, 5), 2414183, 6]-NRT-code), using
- appending kth column [i] based on linear OOA(337, 482844, F3, 4, 5) (dual of [(482844, 4), 1931339, 6]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(337, 482844, F3, 5) (dual of [482844, 482807, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(337, 531441, F3, 5) (dual of [531441, 531404, 6]-code), using
- an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(337, 531441, F3, 5) (dual of [531441, 531404, 6]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(337, 482844, F3, 5) (dual of [482844, 482807, 6]-code), using
- appending kth column [i] based on linear OOA(337, 482844, F3, 4, 5) (dual of [(482844, 4), 1931339, 6]-NRT-code), using
(32, 32+5, large)-Net in Base 3 — Upper bound on s
There is no (32, 37, large)-net in base 3, because
- 3 times m-reduction [i] would yield (32, 34, large)-net in base 3, but