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A linear code of length n and dimension k is a linear subspace C with dimension k of the vector space where is the finite field with q elements. Such a code is called a q -ary code. If q = 2 or q = 3, the code is described as a binary code, or a ternary code respectively. The vectors in C are called codewords.
Linear Codes In order to de ne codes that we can encode and decode e ciently, we add more structure to the codespace. We shall be mainly interested in linear codes. A linear code of length nover the eld Fis a subspace of Fn. Thus the words of linear code the codespace Fnare vectors, and we often refer to codewords as codevectors. codevectors
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Linear Codes. Linear Code: If C is a vector subspace of Fn. Suppose (c1; : : : ; ck) is a basis of the vector subspace C [n; k; d]F code: A code C that is a vector subspace of Fn, of dimension k, and d(C) = d The generator matrix G of a code C is defined to a matrix in. n. as defined below. 0 1. c1.
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Learn the basics of error-correcting codes, from Shannon's information theory to Hamming's combinatorial approach. Explore the minimum distance, decoding algorithms, and applications of linear codes.
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Jun 5, 2012 · An ( n, M, d) code C over a field F = GF ( q) is called linear if C is a linear subspace of Fn over F; namely, for every two codewords c 1, c 2 ∈ C and two scalars a1, a2 ∈ F we have a1 c 1 + a2 c 2 ∈ C. Type. Chapter. Information. Introduction to Coding Theory , pp. 26 - 49.
Nov 23, 2014 · This video is a brief introduction to linear codes: dimensions, G (generating matrix), H (parity check matrix), their forms. Also gives an example of how to ...
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Linear Codes. From Example \(8.16\), it is now easy to check that the minimum nonzero weight is \(3\text{;}\) hence, the code does indeed detect and correct all single errors. We have now reduced the problem of finding “good” codes to that of generating group codes. One easy way to generate group codes is to employ a bit of matrix theory.
