Erasure code

In coding theory, an erasure code is a forward error correction (FEC) code under the assumption of bit erasures (rather than bit errors), which transforms a message of k symbols into a longer message (code word) with n symbols such that the original message can be recovered from a subset of the n symbols. The fraction r = k/n is called the code rate. The fraction k’/k, where k’ denotes the number of symbols required for recovery, is called reception efficiency. In coding theory, an erasure code is a forward error correction (FEC) code under the assumption of bit erasures (rather than bit errors), which transforms a message of k symbols into a longer message (code word) with n symbols such that the original message can be recovered from a subset of the n symbols. The fraction r = k/n is called the code rate. The fraction k’/k, where k’ denotes the number of symbols required for recovery, is called reception efficiency. Optimal erasure codes have the property that any k out of the n code word symbols are sufficient to recover the original message (i.e., they have optimal reception efficiency). Optimal erasure codes are maximum distance separable codes (MDS codes). Parity check is the special case where n = k + 1. From a set of k values { v i } 1 ≤ i ≤ k {displaystyle {v_{i}}_{1leq ileq k}} , a checksum is computed and appended to the k source values: The set of k + 1 values { v i } 1 ≤ i ≤ k + 1 {displaystyle {v_{i}}_{1leq ileq k+1}} is now consistent with regard to the checksum.If one of these values, v e {displaystyle v_{e}} , is erased, it can be easily recovered by summing the remaining variables: In the simple case where k = 2, redundancy symbols may be created by sampling different points along the line between the two original symbols. This is pictured with a simple example, called err-mail: Alice wants to send her telephone number (555629) to Bob using err-mail. Err-mail works just like e-mail, except