A Family of Low-Complexity Binary Codes with Constant Hamming Weights

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^ℓ, M=2^k, d=2) constant-weight codes C[ℓ, r] parameterized by integers ℓ≥ 3 and 1 ≤ r ≤⌊ℓ+3/4⌋, by encoding information in the gaps between successive 1's of a vector. The code has weight w = ℓ and combinatorial dimension k that scales quadratically with ℓ. 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 ⌊log_2 n w⌋. When ℓ =3, the code is optimal achieving the upper bound; when ℓ=4, it is one bit away from the upper bound, and as ℓ grows it is order-optimal in the sense that the ratio of k with its upper bound becomes a constant 11/16 when r=⌊ℓ+3/4⌋. With the same or even lower complexity, we derive new codes permitting a wider range of parameters by modifying C[ℓ, r] in two different ways. The code derived using the first approach has the same blocklength n=2^ℓ, but weight w is allowed to vary from ℓ-1 to 1. In the second approach, the weight remains fixed as w = ℓ, but the blocklength is reduced to n=2^ℓ - 2^r +1. For certain selected values of parameters, these modified codes have an optimal k.