Limiting efficiencies of burst-correcting array codes

The author evaluates the limiting efficiencies e(-S ) of burst-correcting array codes A(n₁,n₂, -s) for all negative readouts -s as n₂ tends to infinity and n₁ is properly chosen to maximize the efficiency. Specializing the result to the products of the first i primes donated by s_i (1⩽i<∞), which are optimal choices for readouts, gives the expression e(-s_i)=(2p_i+1 -2)/(2p_i+1-1) where p_i+1 is the next prime. Previously, it was known only that e(-2)⩾4/5 and e(-1)⩾2/3. This result reveals the existence of burst-correcting array codes with efficiencies arbitrarily close to 1 and with rates also arbitrarily close to 1