Best Known (56, 69, s)-Nets in Base 4
(56, 69, 2734)-Net over F4 — Constructive and digital
Digital (56, 69, 2734)-net over F4, using
- net defined by OOA [i] based on linear OOA(469, 2734, F4, 13, 13) (dual of [(2734, 13), 35473, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(469, 16405, F4, 13) (dual of [16405, 16336, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(469, 16410, F4, 13) (dual of [16410, 16341, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- linear OA(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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(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(12) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(469, 16410, F4, 13) (dual of [16410, 16341, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(469, 16405, F4, 13) (dual of [16405, 16336, 14]-code), using
(56, 69, 8616)-Net over F4 — Digital
Digital (56, 69, 8616)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(469, 8616, F4, 13) (dual of [8616, 8547, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(469, 16410, F4, 13) (dual of [16410, 16341, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(8) [i] based on
- linear OA(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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(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(12) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(469, 16410, F4, 13) (dual of [16410, 16341, 14]-code), using
(56, 69, 6644267)-Net in Base 4 — Upper bound on s
There is no (56, 69, 6644268)-net in base 4, because
- 1 times m-reduction [i] would yield (56, 68, 6644268)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 87112 333302 142438 129752 878880 666412 480870 > 468 [i]