Step1/2

(a) To show that g(x) is a valid generator polynomial, we need to verify that it has the following properties:

g(x) divides x^15 - 1 (the polynomial representing all possible binary sequences of length 15)

g(x) has no factors in common with x^15 - 1 / g(x)

First, we can factor x^15 - 1 over the field of binary numbers as follows: x^15 - 1 = (x^5 - 1)(x^10 + x^5 + 1) = (x - 1)(x^4 + x^3 + x^2 + x + 1)(x^10 + x^5 + 1)

Note that x^5 - 1 = (x - 1)(x^4 + x^3 + x^2 + x + 1), which is the irreducible factorization of x^5 - 1 over the field of binary numbers.

Next, we can verify that g(x) divides x^15 - 1: x^15 - 1 = (x^5 - 1)(x^10 + x^5 + 1) = (x - 1)(x^4 + x^3 + x^2 + x + 1)(x^10 + x^5 + 1) g(x) = x^5 + x^4 + x^2 + 1 g(x) divides x^5 - 1, since (x - 1) is a factor of x^5 - 1, and g(1) = 1 + 1 + 1 + 1 = 0, so (x - 1) is a factor of g(x). Therefore, g(x) divides x^15 - 1.

Finally, we need to show that g(x) has no factors in common with x^15 - 1 / g(x). We can use Euclid's algorithm to compute the greatest common divisor (gcd) of x^15 - 1 and g(x):

(x^15 - 1, g(x)) = (x^10 + x^5 + 1, x^5 + x^4 + x^2 + 1) = (x^10 + x^5 + 1 - x^5(x^5 + x^4 + x^2 + 1), x^5 + x^4 + x^2 + 1) = (x^10 + x^4 + x^2 + 1, x^5 + x^4 + x^2 + 1) = (x^5(x^5 + x^4 + x^2 + 1) + x^4 + x^2, x^5 + x^4 + x^2 + 1) = (x^4 + x^2 + 1, x^5 + x^4 + x^2 + 1) = (x^5 + x^4 + x^2 + 1 - x(x^4 + x^2 + 1), x^4 + x^2 + 1) = (x^3 + x^2 + 1, x^4 + x^2 + 1) = (x^4 + x^2 + 1 - x(x^3 + x^2 + 1), x^3 + x^2 + 1) = (x^2 + x + 1, x^3 + x^2 +1)

Since the gcd is 1, g(x) has no factors in common with x^15 - 1 / g(x), and therefore g(x) is a valid generator polynomial for the binary cyclic code of length 15.

(b) To compute the parity check polynomial for C, we can use the fact that the parity check polynomial is the reciprocal of the generator polynomial modulo x^15 - 1. That is, if the generator polynomial is g(x), then the parity check polynomial is h(x) = x^r g(x^-1) mod (x^15 - 1), where r is the degree of the generator polynomial.

In this case, the degree of the generator polynomial g(x) is 5, so r = 15 - 5 = 10. We can compute g(x^-1) as follows:

g(x^-1) = x^-5 + x^-6 + x^-8 + 1

To find the reciprocal of g(x^-1) modulo x^15 - 1, we need to compute the inverse of each coefficient modulo 2 (since we are working with a binary field). For example, the inverse of 1 modulo 2 is 1, and the inverse of x^-5 = 1/x^5 modulo 2 is x^10, since (x^-5)(x^10) = 1 modulo x^15 - 1.

Using this method, we can compute:

g(x^-1) = x^-5 + x^-6 + x^-8 + 1 = x^10 + x^9 + x^7 + 1

Now we can compute h(x) as:

h(x) = x^r g(x^-1) mod (x^15 - 1) = x^10 (x^10 + x^9 + x^7 + 1) mod (x^15 - 1) = x^10 + x^9 + x^6 + x^4 + x^3 + x^2

Therefore, the parity check polynomial for C is h(x) = x^10 + x^9 + x^6 + x^4 + x^3 + x^2.

(c) The dimension of C is given by k = n - deg(g(x)) = 15 - 5 = 10. Since C is a binary cyclic code, we know that the number of codewords is 2^k, so there are 2^10 = 1024 codewords in C.

(d) To construct the generator matrix for C, we can use the fact that the generator matrix for a binary cyclic code with generator polynomial g(x) is given by G = [g(A), g(A)x, g(A)x^2, ..., g(A)x^(k-1)], where A is the companion matrix of g(x) and k is the dimension of the code.

The companion matrix of g(x) is given by:

A = [0 0 0 0 0 1; 1 0 0 0 0 1; 0 1 0 0 0 1; 0 0 1 0 0 1; 0 0 0 1 0 1]

Therefore, the generator matrix for C is:

G = [g(A) g(A)x g(A)x^2 g(A)x^3 g(A)x^4 g(A)x^5 g(A)x^6 g(A)x^7 g(A)x^8 g(A)x^9; g(A)x^10 g(A)x^11 g(A)x ^10 g(A)x^11 g(A)x^12 g(A)x^13 g(A)x^14 0 0 0 0; 0 g(A) g(A)x g(A)x^2 g(A)x^3 g(A)x^4 g(A)x^5 g(A)x^6 g(A)x^7 g(A)x^8; 0 0 g(A) g(A)x g(A)x^2 g(A)x^3 g(A)x^4 g(A)x^5 g(A)x^6 g(A)x^7; 0 0 0 g(A) g(A)x g(A)x^2 g(A)x^3 g(A)x^4 g(A)x^5 g(A)x^6; 0 0 0 0 g(A) g(A)x g(A)x^2 g(A)x^3 g(A)x^4 g(A)x^5; 0 0 0 0 0 g(A) g(A)x g(A)x^2 g(A)x^3 g(A)x^4; 0 0 0 0 0 0 g(A) g(A)x g(A)x^2 g(A)x^3; 0 0 0 0 0 0 0 g(A) g(A)x g(A)x^2; 0 0 0 0 0 0 0 0 g(A) g(A)x]

To construct the parity check matrix for C, we can use the fact that the parity check matrix for a binary cyclic code with parity check polynomial h(x) is given by H = [-h(A^-1), I_{n-k}], where A^-1 is the inverse of the companion matrix of h(x), I_{n-k} is the (n-k) x (n-k) identity matrix, and k is the dimension of the code.

To find the inverse of A, we first compute the characteristic polynomial of A:

p(x) = det(xI - A) = x^6 + x^5 + x^4 + x^3 + x + 1

Since A is invertible, p(x) is a unit in the polynomial ring Z_2[x], so we can use the extended Euclidean algorithm to find polynomials u(x) and v(x) such that up(x) + vp(x)(x^6 + x^5 + x^4 + x^3 + x + 1) = 1. We get:

u(x) = x^4 + x^3 + x^2 + x + 1 v(x) = x^5 + 1

Therefore, the inverse of A is:

A^-1 = v(A)p(A)^-1 mod 2 = (A^5 + 1)(A^6 + A^5 + A^4 + A^3 + A + 1) mod 2

Using this formula, we can compute A^-1:

A^-1 = [0 0 0 0 1 1; 1 0 0 0 0 1; 1 1 0 0 0 1; 1 1 1 0 0 1; 1 1 1 0 1; 1 1 0 1 0 1; 1 1 1 0 1 1; 0 1 1 1 1 1; 0 0 1 1 1 0; 0 0 0 1 1 1]

To verify that this is the correct parity check matrix, we can multiply it by the transpose of the generator matrix:

G^T = [1 0 0 0 1 1 0 1 0 0 0 0 0 0 0; 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0; 0 0 1 0 0 0 1 1 0 1 0 0 0 0 0; 0 0 0 1 0 0 0 1 1 0 1 0 0 0 0; 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0; 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0; 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0; 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1; 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0; 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1; 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1; 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0; 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0; 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0; 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1]

H^T * G^T = [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;

As expected, the product of the parity check matrix and the transpose of the generator matrix yields the zero matrix, indicating that all codewords of the cyclic code C are orthogonal to the parity check matrix H.

(d) The generator matrix for the binary cyclic code C of length 15 generated by g(x) = x^5 + x^4 + x^2 + 1 is given by:

G = [1 0 0 0 1 1 0 1 0 0 0 0 0 0 0; 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0; 0 0 1 0 0 0 1 1 0 1 0 0 0 0 0; 0 0 0 1 0 0 0 1 1 0 1 0 0 0 0; 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0; 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0; 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0; 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1; 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0; 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1; 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1]

The parity check matrix for the binary cyclic code C is:

H = [1 1 1 0 1 0 0 0 0 0 0 0 0 0 0; 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0; 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0; 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0; 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0; 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0; 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0; 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0; 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0; 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0;

Step2/2

To determine the dimension of the binary cyclic code C, we need to find the rank of the generator matrix G. Using row operations, we can reduce G to the following row echelon form:

G' = [1 0 0 0 1 1 0 1 0 0 0 0 0 0 0; 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0; 0 0 1 0 0 0 1 1 0 1 0 0 0 0 0; 0 0 0 1 0 0 0 1 1 0 1 0 0 0 0; 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0; 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0; 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0; 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1; 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0; 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1; 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0]

The rank of G is 11, so the dimension of C is 15 - 11 = 4.

To find the number of codewords in C, we use the fact that the number of codewords in a binary cyclic code of length n and dimension k is 2^k. Therefore, the number of codewords in C is 2^4 = 16.

Therefore, the binary cyclic code C of length 15 generated by g(x) = x^5 + x^4 + x^2 + 1 has dimension 4 and 16 codewords.

The generator and parity check matrices for C are:

G = [1 0 0 0 1 1 0 1 0 0 0 0 0 0 0; 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0; 0 0 1 0 0 0 1 1 0 1 0 0 0 0 0; 0 0 0 1 0 0 0 1 1 0 1 0 0 0 0]

H = [1 1 1 0 1 0 0 0 0 0 0 0 0 0 0; 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0; 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0; 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0;

The parity check polynomial for C is h(x) = x^10 + x^9 + x^7 + x^6 + x^3 + 1.

To construct the generator matrix for C, we first write the powers of α, the primitive element of GF(2^4), in the columns of the matrix:

G = [1 α^5 α^10 α^15 α^20 α^25 α^30 α^35 α^40 α^45 α^50 α^55 α^60 α^65 α^70; 0 1 α^5 α^10 α^15 α^20 α^25 α^30 α^35 α^40 α^45 α^50 α^55 α^60 α^65; 0 0 1 α^5 α^10 α^15 α^20 α^25 α^30 α^35 α^40 α^45 α^50 α^55 α^60; 0 0 0 1 α^5 α^10 α^15 α^20 α^25 α^30 α^35 α^40 α^45 α^50 α^55]

Next, we multiply G by the inverse of the row echelon form of H to obtain the reduced row echelon form of the parity check matrix H. The reduced row echelon form of H is:

H' = [1 0 0 0 0 0 0 0 1 1 1 1 0 0 0; 0 1 0 0 0 0 0 0 1 1 0 0 1 1 0; 0 0 1 0 0 0 0 0 1 0 1 0 1 0 1; 0 0 0 1 0 0 0 0 0 1 1 0 1 1 1]

The last four columns of H' are the negations of the first four columns of G. Therefore, the generator matrix for C is:

G = [1 0 0 0 1 α^5 α^10 α^15 α^20 α^25 α^30 α^35 α^40 α^45 α^50; 0 1 0 0 0 1 α^5 α^10 α^15 α^20 α^25 α^30 α^35 α^40 α^45; 0 0 1 0 0 0 1 α^5 α^10 α^15 α^20 α^25 α^30 α^35 α^40; 0 0 0 1 0 0 0 1 α^5 α^10 α^15 α^20 α^25 α^30 α^35]

To construct the parity check matrix for C, we can use the parity check polynomial h(x) and the powers of α, the primitive element of GF(2^4), to form the matrix:

H = [h(α^0) h(α^1) h(α^2) h(α^3) h(α^4) h(α^5) h(α^6) h(α^7) h(α^8) h(α^9) h(α^10) h(α^11) h(α^12) h(α^13) h(α^14)]

Substituting h(x) = x^10 + x^9 + x^7 + x^6 + x^3 + 1 and evaluating at the powers of α, we get:

H = [1 0 0 0 0 1 1 0 0 1 1 0 1 0 0; 0 1 0 0 0 1 0 1 0 1 0 1 0 1 0; 0 0 1 0 0 0 1 1 0 0 1 1 0 0 1; 0 0 0 1 0 1 1 1 0 1 1 0 0 1 1]

The generator matrix and parity check matrix for C can be used to encode and decode messages transmitted over the code. The dimension of C is 10, which means that there are 2^10 = 1024 codewords in C.

One important property of cyclic codes is that they can be efficiently encoded and decoded using a cyclic shift register. The generator matrix G can be used to construct a shift register for encoding messages. The shift register has 15 stages, with the feedback connections corresponding to the nonzero coefficients of G. The message bits are shifted into the shift register from left to right, and the output bits are read out from right to left. For example, to encode the message m = (1 0 1 1), we first write it as a column vector and pad it with five zeros to obtain the vector m' = (1 0 1 1 0 0 0 0 0 0 0 0 0 0 0). We then multiply m' by G to obtain the codeword c = (1 0 1 1 1 0 0 0 1 1 1 0 1 0 0). To decode a received codeword r, we can use the parity check matrix H to determine if there are any errors in the received message. The received message is first written as a column vector and multiplied by H to obtain the syndrome vector s. If the syndrome vector is zero, then the received message is a valid codeword. Otherwise, the syndrome vector can be used to determine the location of the error in the received message, and the error can be corrected using the algebraic properties of cyclic codes. In general, the minimum distance of a cyclic code is related to the weight distribution of its generator polynomial. The generator polynomial for C has three nonzero terms, so the code can have at most three nonzero coefficients in each codeword. The minimum distance of C is therefore either 1, 2, or 3. Since C is a binary code, it cannot have a minimum distance of 1, so the minimum distance of C is either 2 or 3. The parity check matrix for C can be used to calculate the weight distribution of the code and determine its minimum distance.

Explanation:

(a) We showed that g(x) = x^5 + x^4 + x^2 + 1 is a valid generator polynomial for the binary cyclic code C of length 15.

(b) We computed the parity check polynomial h(x) for C using the fact that g(x)h(x) = x^15 + 1. We obtained h(x) = x^10 + x^9 + x^8 + x^6 + x^5 + x^3 + x^2 + 1.

(c) We determined the dimension and the number of codewords of C. We found that the dimension of C is 10, since the degree of g(x) is 5 and the length of C is 15, so the number of information bits is 10. We also found that the number of codewords of C is 2^10 = 1024, since C is a linear code over GF(2).

(d) We constructed the generator matrix G and the parity check matrix H for C using the generator and parity check polynomials g(x) and h(x), respectively. We obtained:

G = [1 0 0 0 0 1 0 0 1 0 0 0 1 0 1] [0 1 0 0 0 0 1 0 0 1 0 0 1 0 1] [0 0 1 0 0 0 0 1 0 0 1 0 1 0 1] [0 0 0 1 0 0 0 0 1 0 0 1 0 1 1]

H = [1 0 1 1 1 0 0 0 1 1 1 0 1 0 0] [0 1 1 0 1 1 0 0 1 1 0 1 0 1 0] [0 0 1 1 0 1 1 0 1 0 1 1 0 1 1] [0 0 0 1 1 0 1 1 0 1 1 1 1 1 0]

Public Answer