We define and study burst-covering codes. We provide some general bounds connecting the parameters of a code with its burst-covering radius. We then provide stronger bounds on the burst-covering radius of cyclic codes, by employing linear-feedback shift-register (LFSR) sequences. For the case of BCH codes we prove a bound on pattern frequencies in LFSR sequences, and using this tool, we bound the burst-covering radius of binary primitive BCH codes and Melas codes. We then present an efficient burst-covering algorithm for cyclic codes. Finally, we present a bound on the critical exponent of cyclic codes based on the burst-covering radius.
We prove a new lower bound on the $b$-symbol weight of codewords in dual Melas codes, which is strictly better than known results. We employ a new strategy that translates a pattern counting problem in linear-feedback shift register sequences to a problem on orbits of binary polynomials under a certain action. We use the same techniques to provide an improved lower bound on the burst-covering radius of Melas codes and a new upper bound on the critical exponent of dual Melas codes. We then define a new randomness postulate on (families of) binary sequences, based on the definition of infinite Poisson generic sequences. We conjecture that codewords in the dual Melas code satisfy this postulate, and we show partial progress in establishing this conjecture. We also show how a relaxed version of the problem can be completely resolved, and relate the conjecture to the first part of the paper, giving a justification of why the improved bounds on the $b$-symbol weight were somewhat to be expected.