P17125 [ICPC 2025 Shanghai R] Flower' s land 3

There is a binary string $s_1$ of length $m$ stored on disk $\#1$. Then, there are $n - 1$ operations. In the $i$-th operation, a disk $1 \le p_{i+1} \le i$ is chosen, and the string $s_{p_{i+1}}$ on that disk is copied to disk $i + 1$, producing $s_{i+1}$. However, during the copying process, up to $k$ bits may be flipped (i.e., errors occur in at most $k$ positions). You are given all the final $n$ binary strings $s_1, s_2, \cdots, s_n$, each of length $m$. Your task is to determine how many possible sequences $p_2, p_3, \cdots, p_n$ could result in this final configuration. Since the answer can be large, you only need to find the answer modulo $998244353$.

The first line of the input contains three integers $n$, $m$, $k$ ($2 \le n \le 5000$, $4 \le m \le 15000$, $1 \le k \le 3$), described in the statement. It is guaranteed that $m$ is a multiple of $4$. Each of the next $n$ lines contains a hexadecimal string $s_i'$ of length $m/4$. Each character of $s_i'$ is one of $0$-$9$ or $A$-$F$, where $A = 10$, $B = 11$, $\cdots$, $F = 15$. Each bit in the hexadecimal representation $s_i'$ corresponds to $4$ consecutive bits in the binary string $s_i$. Specifically, for each bit $s'_{i,j}$, it can be proved that there exists a unique tuple $(a,b,c,d)$ satisfying $s'_{i,j} = 8a + 4b + 2c + d$ and $a,b,c,d \in \{0,1\}$. Bits in the binary string $s_i$ satisfy $(s_{i,4j}, s_{i,4j+1}, s_{i,4j+2}, s_{i,4j+3}) = (a,b,c,d)$.

Print an integer — the number of valid sequences $(p_2, p_3, \cdots, p_n)$ modulo $998244353$.