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CHAPTER 03 - Cyclic codes

# CHAPTER 03 - Cyclic codes - IV054 CHAPTER 3 Cyclic and...

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Cyclic codes 1 CHAPTER 3: CHAPTER 3: Cyclic and convolution codes Cyclic and convolution codes Cyclic codes are of interest and importance because They posses rich algebraic structure that can be utilized in a variety of ways. They have extremely concise specifications. They can be efficiently implemented using simple shift registers . Many practically important codes are cyclic. Convolution codes allow to encode streams od data (bits). IV054

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2 Cyclic codes IMPORTANT NOTE In order to specify a binary code with 2 k codewords of length n one may need to write down 2 k codewords of length n. In order to specify a linear binary code with 2k codewords of length n it is sufficient to write down k codewords of length n. In order to specify a binary cyclic code with 2 k codewords of length n it is sufficient to write down 1 codeword of length n .
3 Cyclic codes BASIC BASIC DEFINITION DEFINITION AND AND EXAMPLES EXAMPLES Definition A code C is cyclic if (i) C is a linear code; (ii) any cyclic shift of a codeword is also a codeword, i.e. whenever a 0 ,… a n -1 C , then also a n -1 a 0 a n –2 C . IV054 Example (i) Code C = {000, 101, 011, 110} is cyclic. (ii) Hamming code Ham (3, 2): with the generator matrix is equivalent to a cyclic code. (iii) The binary linear code {0000, 1001, 0110, 1111} is not a cyclic, but it is equivalent to a cyclic code . (iv) Is Hamming code Ham (2, 3) with the generator matrix (a) cyclic? (b) equivalent to a cyclic code? = 1 1 1 1 0 0 0 0 1 1 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 0 1 G 2 1 1 0 1 1 0 1

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4 Cyclic codes F F REQUENCY of CYCLIC CODES REQUENCY of CYCLIC CODES Comparing with linear codes, the cyclic codes are quite scarce. For, example there are 11 811 linear (7,3) linear binary codes, but only two of them are cyclic. Trivial cyclic codes . For any field F and any integer n >= 3 there are always the following cyclic codes of length n over F : No-information code - code consisting of just one all-zero codeword. Repetition code - code consisting of codewords ( a , a , …, a ) for a F . Single-parity-check code - code consisting of all codewords with parity 0. No-parity code - code consisting of all codewords of length n For some cases, for example for n = 19 and F = GF (2), the above four trivial cyclic codes are the only cyclic codes. IV054
5 Cyclic codes EXAMPLE of a CYCLIC CODE EXAMPLE of a CYCLIC CODE The code with the generator matrix has codewords c 1 = 1011100 c 2 = 0101110 c 3 =0010111 c 1 + c 2 = 1110010 c 1 + c 3 = 1001011 c 2 + c 3 = 0111001 c 1 + c 2 + c 3 = 1100101 and it is cyclic because the right shifts have the following impacts c 1 c 2 , c 2 c 3 , c 3 c 1 + c 3 c 1 + c 2 c 2 + c 3 , c 1 + c 3 c 1 + c 2 + c 3 , c 2 + c 3 c 1 c 1 + c 2 + c 3 c 1 + c 2 IV054 = 1 1 1 0 1 0 0 0 1 1 1 0 1 0 0 0 1 1 1 0 1 G

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6 Cyclic codes POLYNOMIALS POLYNOMIALS over over GF( GF( q q ) ) A
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CHAPTER 03 - Cyclic codes - IV054 CHAPTER 3 Cyclic and...

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