Best Known (45, 53, s)-Nets in Base 4
(45, 53, 16394)-Net over F4 — Constructive and digital
Digital (45, 53, 16394)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 10)-net over F4, using
- digital (40, 48, 16384)-net over F4, using
- net defined by OOA [i] based on linear OOA(448, 16384, F4, 8, 8) (dual of [(16384, 8), 131024, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(448, 65536, F4, 8) (dual of [65536, 65488, 9]-code), using
- 1 times truncation [i] based on linear OA(449, 65537, F4, 9) (dual of [65537, 65488, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, 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(449, 65537, F4, 9) (dual of [65537, 65488, 10]-code), using
- OA 4-folding and stacking [i] based on linear OA(448, 65536, F4, 8) (dual of [65536, 65488, 9]-code), using
- net defined by OOA [i] based on linear OOA(448, 16384, F4, 8, 8) (dual of [(16384, 8), 131024, 9]-NRT-code), using
(45, 53, 65564)-Net over F4 — Digital
Digital (45, 53, 65564)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(453, 65564, F4, 8) (dual of [65564, 65511, 9]-code), using
- 1 times truncation [i] based on linear OA(454, 65565, F4, 9) (dual of [65565, 65511, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(4) [i] based on
- linear OA(449, 65536, F4, 9) (dual of [65536, 65487, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(425, 65536, F4, 5) (dual of [65536, 65511, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(45, 29, F4, 3) (dual of [29, 24, 4]-code or 29-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(454, 65565, F4, 9) (dual of [65565, 65511, 10]-code), using
(45, 53, large)-Net in Base 4 — Upper bound on s
There is no (45, 53, large)-net in base 4, because
- 6 times m-reduction [i] would yield (45, 47, large)-net in base 4, but