# lect6 - Error Correcting Codes: Combinatorics, Algorithms...

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Error Correcting Codes: Combinatorics, Algorithms and Applications (Fall 2007) Lecture 6: General Hamming Codes September 10, 2007 Lecturer: Atri Rudra Scribe: Nathan Russell In the last lecture, we saw the following ways of deﬁning an [ n, k, d ] q linear code C : An k × n generator matrix G , so that C is the result of multiplying all vectors x of length n with G , giving codewords C = { x · G | x F k q } . An ( n - k ) × n parity check matrix H such that C = { x F n q | H · x T = 0 } . Note that since x is a row vector, we need to take the transpose so that multiplication is deﬁned. We forgot to explicitly deﬁne the following notion related to linear subspaces in the last lecture. Deﬁnition 0.1 (Linear independence of vectors) . We say that v 1 , v 2 , . . . v k are linearly indepen- dent if for every 1 i k v i 6 = a 1 v 1 + . . . + a i - 1 v i - 1 + a i +1 v i +1 + . . . + a k v k , for every k - 1 -tuple ( a 1 , a 2 , . . . , a i - 1 , a i +1 , . . . , a k ) F k - 1 q . Note that the basis of a linear subspace must be linearly independent.

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## This note was uploaded on 10/16/2011 for the course ELECTRICAL EE251202 taught by Professor Rejaei during the Spring '10 term at Sharif University of Technology.

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lect6 - Error Correcting Codes: Combinatorics, Algorithms...

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