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The form of the parity check matrix of Eq. 49) is called the echelon canonical form. If the parity check matrix of Eq. 49) is used, the check symbols x k+v xk+2> · · ·» xn a r e expressed in terms of the message symbols (wx u2 . . uk) — (xx JC2 . . 50) '••+Pn-k9kUk)· + Any parity check matrix can be transformed into the echelon canonical form by an appropriate transformation. A code generated by the parity check matrix of a code C is called the dual code of C and is denoted by C ± . The dual code of an (n, k) code is an («, n - k) code.

The free distance is given as the minimum weight path in the state diagram which starts at the all-zero state and ends at the all-zero state. 2 Let's consider the binary convolutional code with rate 1/2 and con­ straint length 3 given by the generator matrix G(D) = (1 + Z>2,1 + D + D2).

As well. The minimal polynomial of a primitive element is a primitive polynomial. 43) constitute GF(#). Thus, x«-x= Π (x-ß). ß^GF(q) If M(x) is the minimal polynomial of an element of GF(q\ divisible by M(x). 3. The primitive element a is a zero of f(x) = 1 4- x + x4, and f(x) is the minimal polynomial of a. Hence, we denote ϊ(χ) by M ^ x ) . 5. Minimal Polynomials and Orders of the Elements of GF(2 4 ) Element Minimal Polynomial o NU*) = x - 1 a, a1, a 4 , «8 a 3 ,« 6 , a9, a12 a 5 ,« 1 0 a7, a11, a13, a14 M0(JC) = 1 + x 1 M^JC) = 1 + JC + x4 Order 15 M 3 (JC) = 1 + x + x2 + x 3 4- x4 5 M5(x) = 1 + x + x2 3 M 7 ( J C ) = 1 + JC3 + x4 15 zeros, namely a, a 2 , a 4 , and a8, and M X (JC) is the minimal polynomial for these elements.

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