lect25 - Error Correcting Codes: Combinatorics, Algorithms...

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Unformatted text preview: Error Correcting Codes: Combinatorics, Algorithms and Applications (Fall 2007) Lecture 25: Justesen Codes October 26, 2007 Lecturer: Atri Rudra Scribe: Kanke Gao In the last lecture, we introduced code concatenation, where we compose an outer code C out with an inner code C in . We derived the Zyablov bound by picking C out on the Singleton bound and C in on the GV bound. We also presented a polynomial time construction of a code that achieves the Zyablov bound (and hence, an asymptotically good code). A somewhat unsatisfactory aspect of this construction was the brute force search for a suitable inner code (which lead to the poly- nomial construction time). In todays lecture, we will study a strongly explicit construction of an asymptotically good code. 1 Strongly explicit construction A polynomial time construction of an asymptotically good code was presented in the last lec- ture. A natural followup question is if we can have a strongly explicit construction. Techni- cally speaking, by strongly explicit construction, we mean a log space construction. However, we will not formally define this notation. We will now consider the so called Justesen code [1]. Justesen code is concatenation code with different linear inner codes, which is composed of an ( N,K,D ) q k outer code C out and different ( n,k,d ) q inner codes C i in : 1 i N . Formally, the concatenation of these codes, denoted by...
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lect25 - Error Correcting Codes: Combinatorics, Algorithms...

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