In this paper, we focus on the design of binary constant-weight codes that admit low-complexity encoding and decoding algorithms, and that have size as a power of $2$. We construct a family of $(n=2^\ell, M=2^k, d=2)$ constant-weight codes ${\cal C}[\ell, r]$ parameterized by integers $\ell \geq 3$ and $1 \leq r \leq \lfloor \frac{\ell+3}{4} \rfloor$, by encoding information in the gaps between successive $1$'s of a vector. The code has weight $w = \ell$ and combinatorial dimension $k$ that scales quadratically with $\ell$. The encoding time is linear in the input size $k$, and the decoding time is poly-logarithmic in the input size $n$, discounting the linear time spent on parsing the input. Encoding and decoding algorithms of similar codes known in either information-theoretic or combinatorial literature require computation of large number of binomial coefficients. Our algorithms fully eliminate the need to evaluate binomial coefficients. While the code has a natural price to pay in $k$, it performs fairly well against the information-theoretic upper bound $\lfloor \log_2 {n \choose w} \rfloor$. When $\ell =3$, the code is optimal achieving the upper bound; when $\ell=4$, it is one bit away from the upper bound, and as $\ell$ grows it is order-optimal in the sense that the ratio of $k$ with its upper bound becomes a constant $\frac{11}{16}$ when $r=\lfloor \frac{\ell+3}{4} \rfloor$. With the same or even lower complexity, we derive new codes permitting a wider range of parameters by modifying ${\cal C}[\ell, r]$ in two different ways. The code derived using the first approach has the same blocklength $n=2^\ell$, but weight $w$ is allowed to vary from $\ell-1$ to $1$. In the second approach, the weight remains fixed as $w = \ell$, but the blocklength is reduced to $n=2^\ell - 2^r +1$. For certain selected values of parameters, these modified codes have an optimal $k$.

翻译：本文聚焦于设计具有低复杂度编码与解码算法、且码字数为2的幂次的二元恒定重量码。我们通过将信息编码在向量中连续1之间的间隙上，构造了一个由整数 $\ell \geq 3$ 和 $1 \leq r \leq \lfloor \frac{\ell+3}{4} \rfloor$ 参数化的 $(n=2^\ell, M=2^k, d=2)$ 恒定重量码族 ${\cal C}[\ell, r]$。该码的重量为 $w = \ell$，组合维度 $k$ 随 $\ell$ 呈二次方增长。编码时间与输入规模 $k$ 成线性关系，解码时间（不考虑解析输入所花费的线性时间）与输入规模 $n$ 成多对数关系。信息论或组合文献中已知的类似码的编码与解码算法均需计算大量二项式系数，而我们的算法完全消除了对二项式系数求值的需求。尽管该码在 $k$ 值上存在自然折衷，但其性能相对于信息论上界 $\lfloor \log_2 {n \choose w} \rfloor$ 表现良好。当 $\ell =3$ 时，该码达到上界，为最优码；当 $\ell=4$ 时，与上界仅差1比特；随着 $\ell$ 增长，当 $r=\lfloor \frac{\ell+3}{4} \rfloor$ 时，$k$ 与其上界之比趋于常数 $\frac{11}{16}$，具有阶数最优性。在相同或更低复杂度下，我们通过两种不同方式修改 ${\cal C}[\ell, r]$，推导出能支持更广泛参数的新码。第一种方法得到的码具有相同的码长 $n=2^\ell$，但重量 $w$ 可在 $\ell-1$ 到 $1$ 之间变化。第二种方法中，重量保持为 $w = \ell$，但码长缩减为 $n=2^\ell - 2^r +1$。对于某些特定参数选择，这些修改后的码具有最优的 $k$ 值。