Best Known (69, 69+13, s)-Nets in Base 4
(69, 69+13, 43690)-Net over F4 — Constructive and digital
Digital (69, 82, 43690)-net over F4, using
- net defined by OOA [i] based on linear OOA(482, 43690, F4, 13, 13) (dual of [(43690, 13), 567888, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(482, 262141, F4, 13) (dual of [262141, 262059, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(482, 262144, F4, 13) (dual of [262144, 262062, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(482, 262144, F4, 13) (dual of [262144, 262062, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(482, 262141, F4, 13) (dual of [262141, 262059, 14]-code), using
(69, 69+13, 98924)-Net over F4 — Digital
Digital (69, 82, 98924)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(482, 98924, F4, 2, 13) (dual of [(98924, 2), 197766, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(482, 131072, F4, 2, 13) (dual of [(131072, 2), 262062, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(482, 262144, F4, 13) (dual of [262144, 262062, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- OOA 2-folding [i] based on linear OA(482, 262144, F4, 13) (dual of [262144, 262062, 14]-code), using
- discarding factors / shortening the dual code based on linear OOA(482, 131072, F4, 2, 13) (dual of [(131072, 2), 262062, 14]-NRT-code), using
(69, 69+13, large)-Net in Base 4 — Upper bound on s
There is no (69, 82, large)-net in base 4, because
- 11 times m-reduction [i] would yield (69, 71, large)-net in base 4, but