Lecture 10 : Justesen Codes and Reed - Solomon Decoding 1 November 2006
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چکیده
Up until now, we have seen bounds that tell us good codes should be possible but we didn’t know how to construct such codes explicitly. The binary Reed-Solomon code gave us good rate but poor distance. Intuitively, we should be able to concatenate it with an inner code that would “amplify” its distance. Enumerating and searching codes takes O(2) time, so we can’t just find one good inner code and use it for all outer blocks. A key insight of the Justesen code is the use of different inner codes from an easy-to-find ensemble, most of which have good distance. The Justesen code concatenates Reed-Solomon on the outside with an ensemble of codes over a field of 2 elements on the inside. We denote n as the number of non-zero field elements, 2 − 1, and therefore each field element can be encoded into binary withm = O(log n) bits. The associated Reed-Solomon polynomial ismk-bits long:
منابع مشابه
Lecture 8
So far, we have examined three types of codes: • Atomic Codes: Hamming, Hadamard, Reed-Solomon, and Reed-Muller codes • Random Codes • Concatenated Codes: Forney and Justesen Codes The last two types of codes are asymptotically good. However, while the Random Codes are nonconstructible, the Concatenated Codes provide us with an explicit way of producing asymptotically good codes. Further in the...
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