Question

A (8.4) cyclic code is generated by g(x)=x 4 +x+1. Develop generator and parity check matrix in
systematic​

Answer 1

Answer:

Channel Decoding, 2010

8.4.1 Exact Computation

Assuming that the CRC is obtained from the systematic generator matrix G = [I, Π], it can be evaluated iteratively over the data r, as shown by (8.1). More precisely, the value of the CRC associated to the j + 1 first bits of r (shortly, at time j + 1) only depends on the value of the CRC at time j and on the j + 1 th bit of r. Each value of the CRC at time j leads to two different values of the CRC at time j + 1. Consequently, the evolution of the CRC values according to the bits of r can be described by a trellis. In this trellis, states correspond to the 2ℓ(c) possible values of the CRC. Transitions are determined by the bits of r. At each time j = 1,…, ℓ(r), we study the contribution of rj (the jth bit of r) over the global CRC, see Example 8.1.

Example 8.1

The trellis that may be obtained for a systematic binary Hamming (7,4) code with

is represented in Figure 8.12. In the example, it is assumed that [k, u] = [1] and that ℓ(o) = 3.

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Figure 8.12. Trellis obtained by forward construction, the dotted lines correspond to transitions associated to ci = 0, the plain lines to ci = 1.

Explanation:

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