Information Theory and Coding > Error Control Coding > What is Cyclic Code?

Cyclic Code

Cyclic codes are known to be a subcategory of linear coding technique because these offers efficient encoding and decoding schemes using a shift register.  A code C is called a cyclic if the following two properties satisfies by it:

• C is a linear code, and,
• any cyclic shift of a codeword is also a codeword, i.e., if the codeword a₀a₁•• a_n-1 is in C then a_n-1 a₀ a₁ ••• a_n-2 is also in C.

Explanation:

• Property of Linearity

A linear combination of two codewords must be another codeword. Suppose we have two codewords C_i and C_j. So, on adding C_i + C_j = C_p where this C_p must also be a codeword. For example, suppose we have given 3 codewords (110, 101, 011). the addition of any of the two given codewords must produce the third codeword. i.e 110 + 101 = 011

 

• Property of Cyclic Shifting

After a right or left shift in the bits of codewords the resultant code generated must be another codeword. For example, consider again those 3 codewords (110, 101, 011), an either right or left shift in the bits of a codeword must generate another codeword.

110: shifting the bits towards the right will provide 011.

 

Generator polynomial for Cyclic Codes

For (n.k) binary cyclic code the generator polynomial is

 

Let we have cyclic code (n,k) as

C=(C₀, C₁, C₂,  …. C_n-1)

So  polynomial code is

Consider the first non-zero codeword for (7,4) code C₁ =(1101000)

C₁(x) = 1 + x  + x³

This the generator for (7, 4) hamming code and from this code we can get all the codewords

C₂(x) = x * C₁(x) = x + x² + x⁴       and C₂ =(0110100)

C₃(x) = x * C₂(x) = x² + x³ + x⁵       and C₃ =(0011010)

C₄(x) = x * C₃(x) = x³ + x⁴ + x⁶       and C₄ =(0001101)

 

 

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Encoding the cyclic Code: Non-Systematic Encoding

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