Locally Balanced Constraints

Three new constraints are introduced in this paper. These constraints are characterized by limitations on the Hamming weight of every subword of some fixed even length ℓ. In the (ℓ, δ)-locally-balanced constraint, the Hamming weight of every length-ℓ subword is bounded between ℓ/2 — δ and ℓ/2 + δ. The strong-(ℓ,δ)-locally-balanced constraint imposes the locally- balanced constraint for any subword whose length is at least ℓ. Lastly, the Hamming weight of every length-ℓ subword which satisfies the (ℓ, δ)-locally-bounded constraint is at most ℓ/2 — δ. It is shown that the capacity of the strong-(ℓ, δ)-locally-balanced constraint does not depend on the value of ℓ and is identical to the capacity of the (2δ + 1)-RDS constraint. The latter constraint limits the difference between the number of zeros and ones in every prefix of the word to be at most 2δ + 1. This value is also a lower bound on the capacity of the (ℓ, δ)-locally-balanced constraint, while a corresponding upper bound is given as well. Lastly, it is shown that if δ is not large enough, namely for $\delta < \sqrt \ell /2$, then the capacity of the (ℓ, δ)-locally-bounded constraint approaches 1 as ℓ increases.