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UNIT III

UNIT III. Linear Block Codes. A linear code is an important type of block code used in error correction and detection schemes. Linear codes allow for more efficient encoding and decoding algorithms than other codes (cf. syndrome decoding).

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UNIT III

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  1. UNIT III

  2. Linear Block Codes • A linear code is an important type of block code used in error correction and detection schemes. Linear codes allow for more efficient encoding and decoding algorithms than other codes (cf. syndrome decoding). • Linear codes are applied in methods of transmitting symbols (e.g., bits) on a communications channel so that, if errors occur in the communication, some errors can be detected by the recipient of a message block. The "codes" in the linear code are blocks of symbols which are encoded using more symbols than the original value to be sent. A linear code of length n transmits blocks containing n symbols.

  3. Syndrome decoding We construct a Hamming code C, encode an information word using C, introduce one error, and then decode by calculating the syndrome of the ``received'' vector and applying the CosetLeaders map to the syndrome to recover the original vector.

  4. Cyclic Codes • cyclic codes find an important application in error detection and correction. • Let C be a linear code over a finite field A of block length n. C is called a cyclic code, if for every codeword c=(c1,...,cn) from C, the word (cn,c1,...,cn-1) in An obtained by a cyclic right shift of components is also a codeword from C. • Sometimes, C is called the c-cyclic code, if C is the smallest cyclic code containing c, or, in other words, C is the linear code generated by c and all codewords obtained by cyclic shifts of its components. • Cyclic codes can be linked to ideals in certain rings. Let R = A[x] / (xn − 1). Identify the elements of the cyclic code C with polynomials in R such that maps to the polynomial : • thus multiplication by x corresponds to a cyclic shift. Then C is an ideal in R, and hence principal, since R is a principal ideal ring. The ideal is generated by the unique element in C of minimum degree, the generator polynomial g. [1] This must be a divisor of xn − 1. If the generator polynomial g has degree d then the the rank of the code C is n − d. • For example, if A= and n=3, the codewords contained in the (1,1,0)-cyclic code are precisely • (0,0,0),(1,1,0),(0,1,1) and (1,0,1). • It corresponds to the ideal in generated by (1 + x).

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