Best Known (39, 39+8, s)-Nets in Base 4
(39, 39+8, 4106)-Net over F4 — Constructive and digital
Digital (39, 47, 4106)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 10)-net over F4, using
- digital (34, 42, 4096)-net over F4, using
- net defined by OOA [i] based on linear OOA(442, 4096, F4, 8, 8) (dual of [(4096, 8), 32726, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(442, 16384, F4, 8) (dual of [16384, 16342, 9]-code), using
- 1 times truncation [i] based on linear OA(443, 16385, F4, 9) (dual of [16385, 16342, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(443, 16385, F4, 9) (dual of [16385, 16342, 10]-code), using
- OA 4-folding and stacking [i] based on linear OA(442, 16384, F4, 8) (dual of [16384, 16342, 9]-code), using
- net defined by OOA [i] based on linear OOA(442, 4096, F4, 8, 8) (dual of [(4096, 8), 32726, 9]-NRT-code), using
(39, 39+8, 16409)-Net over F4 — Digital
Digital (39, 47, 16409)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(447, 16409, F4, 8) (dual of [16409, 16362, 9]-code), using
- 1 times truncation [i] based on linear OA(448, 16410, F4, 9) (dual of [16410, 16362, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(4) [i] based on
- linear OA(443, 16384, F4, 9) (dual of [16384, 16341, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(422, 16384, F4, 5) (dual of [16384, 16362, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(45, 26, F4, 3) (dual of [26, 21, 4]-code or 26-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(8) ⊂ Ce(4) [i] based on
- 1 times truncation [i] based on linear OA(448, 16410, F4, 9) (dual of [16410, 16362, 10]-code), using
(39, 39+8, large)-Net in Base 4 — Upper bound on s
There is no (39, 47, large)-net in base 4, because
- 6 times m-reduction [i] would yield (39, 41, large)-net in base 4, but