On Correcting Bursts (and Random Errors) in Vector Symbol (n, k) Cyclic Codes

摘要

In this communication, simple methods are shown for correcting bursts of large size and bursts combined with random errors using vector symbols and primarily vector xor and feedback shift register operations. One result is that any $(n, k)$ cyclic code with minimum distance ${> 2}$ can correct all full vector symbol error bursts of length $n-k-1$ or less if the error vectors are linearly independent. If the bursts are not full but contain some error-free components, the capability of correcting bursts up to $n-k$ or less is code dependent. Also, vector symbol decoding with Reed–Solomon component codes can correct, very simply, with probability ${geq 1}-n(n-k)2 ^{ -r}$, all cases of $e leq n-k-1~r$-bit random errors in any cyclic span of length $leq n-k$. The techniques often work when there is linear dependence. In cases where most errors are in a burst but a small number of errors are outside, the solution, given error-correcting capability, can be broken down into a simple solution for the small number of outside errors, followed by a simple subtraction to reveal all the error values in the burst part.