Two Theorems in List Decoding

We prove the following results concerning the list decoding of error-correcting codes: (i) We show that for \textit{any} code with a relative distance of $\delta$ (over a large enough alphabet), the following result holds for \textit{random errors}: With high probability, for a \rho\le \delta -\eps fraction of random errors (for any \eps>0), the received word will have only the transmitted codeword in a Hamming ball of radius $\rho$ around it. Thus, for random errors, one can correct twice the number of errors uniquely correctable from worst-case errors for any code. A variant of our result also gives a simple algorithm to decode Reed-Solomon codes from random errors that, to the best of our knowledge, runs faster than known algorithms for certain ranges of parameters. (ii) We show that concatenated codes can achieve the list decoding capacity for erasures. A similar result for worst-case errors was proven by Guruswami and Rudra (SODA 08), although their result does not directly imply our result. Our results show that a subset of the random ensemble of codes considered by Guruswami and Rudra also achieve the list decoding capacity for erasures. Our proofs employ simple counting and probabilistic arguments.Comment: 19 pages, 0 figure

Catalytic quantum error correction

We develop the theory of entanglement-assisted quantum error correcting (EAQEC) codes, a generalization of the stabilizer formalism to the setting in which the sender and receiver have access to pre-shared entanglement. Conventional stabilizer codes are equivalent to dual-containing symplectic codes. In contrast, EAQEC codes do not require the dual-containing condition, which greatly simplifies their construction. We show how any quaternary classical code can be made into a EAQEC code. In particular, efficient modern codes, like LDPC codes, which attain the Shannon capacity, can be made into EAQEC codes attaining the hashing bound. In a quantum computation setting, EAQEC codes give rise to catalytic quantum codes which maintain a region of inherited noiseless qubits. We also give an alternative construction of EAQEC codes by making classical entanglement assisted codes coherent.Comment: 30 pages, 10 figures. Notation change: [[n,k;c]] instead of [[n,k-c;c]

Codes for protection from synchronization loss and additive errors

Codes for protection from synchronization loss and additive error