Optimal Break-Resilient Codes

Abstract

Break-resilient codes enable reliable communication in the presence of an omniscient adversary that may split a transmitted message at arbitrary boundaries between consecutive symbols, while the receiver observes only an unordered multiset of the resulting fragments. For binary codewords of length~$n$ subject to at most~$t$ breaks, the best known explicit construction has redundancy~$O(t\log_2 n\log_2\log_2\log_2 n)$, whereas the information-theoretic lower bound is~$\Omega(t\log_2(n/t))$. In this paper, we extend the binary break model to any fixed finite field~$\bbF_q$ and establish a redundancy lower bound of~$\Omega(t\log_q(n/t))$. We then give an explicit construction of~$q$-ary break-resilient codes with redundancy~$O(t\log_q n)$ when~$t\leq n^{1-\varepsilon}$ for a fixed constant~$\varepsilon\in(0,1)$, matching the information-theoretic lower bound up to a constant factor. The key idea is to compute a short algebraic fingerprint of the message, which enables the decoder to reject incorrect assemblies of the received fragments.