On Linear and Cyclic Codes for Correcting Deletions

The use of linear and cyclic codes for correcting synchronization errors is investigated. These codes are typically required to have the capability to correct substitution errors, which are the most common type of errors in most communication and storage systems. However, synchronization errors, associated with the deletion and insertion of symbols, can also occur and may cause severe performance degradation unless the coding scheme possesses the capability to recover from such errors. We show that linear codes of rate greater than 1/2 cannot correct deletion or insertion errors. Although cyclic codes, except for repetition codes, cannot correct deletion or insertion errors, we show that adding one extra symbol to the codewords of a binary or nonbinary cyclic code of rate at most 1/3 or 1/2, respectively, yields a code capable of correcting a single deletion or insertion.