MATHEMATICAL BACKGROUND AND MATRIX CODES

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2.4 2.5

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Find all primitive elements in GF 7 and GF 11 . Construct a table for GF 23 de ned by the primitive polynomial p x x3 x 1. Express each element in GF 23 by power, polynomial, vector, and matrix representations. Show that the polynomial x5 x2 1 is irreducible and primitive over GF 2 . Prove that reciprocal polynomial p x is irreducible if and only if p x is irreducible over GF 2 . Also prove that p x is primitive if and only if p x is primitive. Find all irreducible polynomials of degree 2 over GF 3 and determine which of these are primitive. Let the primitive polynomial with degree 5 be p x x5 x2 1, and also a be a root of p x , meaning a primitive element of GF 25 . Find the minimal polynomials of a3 , a5 , and a7 . Let a be a root of primitive polynomial x2 x 2 over GF 3 . Find the minimal polynomials of all elements in GF 32 . Transform the matrices below to systematic forms. 1 6 H1 4 0 a 6 H2 4 b 1 2 1 2 1 1 1 1 0 b 0 1 1 b 1 a 3 1 1 7 1 05 0 1 0 1 a b 1 0

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2.6 2.7 2.8 2.9

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2.10 2.11

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over 3 b 7 a5 a

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GF 2 ;

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over

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GF 4 :

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2.12 2.13

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Prove that the companion matrix is nonsingular. Show that rows or columns of nonsingular matrix are linearly independent. And show that any nonsingular matrix can be transformed into identity matrix by elementary row operations. Construct the vector space of all 3-tuples over GF 3 . Form a two-dimensional subspace and its null space. Given the following H matrix of a linear code C, answer the questions below: 1 H 0 (a) (b) (c) (d) 1 a 1 b b a ! over GF 4 :

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2.14 2.15

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Transform the above matrix to a systematic form. Find all codewords of C. Encode a given message a b . Decode a received word 1 b b 0 .

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Prove that the code is capable of correcting any combinations of random t errors and e erasures if the minimum distance of the code dmin is at least 2t e 1.

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BASIC MATRIX CODES

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Let a linear code be expressed by the parity-check matrix H, 1 H 1 1 a 1 a2 1 0 0 1 ! over GF 4 ;

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where a is a root of the binary polynomial x2 x 1. (a) (b) (c) (d) (e) Show that the set f0; 1; a; a2 g is GF 4 . Show that the code is a single-error correcting code. For the given input information 1 0 a2 , nd the transmitted word. Decode the received word 1 a 0 a2 1 . For the single-error correcting code over GF 4 expressed by a parity-check matrix with r rows, express the maximum code length by using r. Also mention how the answer is deduced.

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Let C be a binary code with code length n and information length k whose paritycheck matrix H is organized by distinct odd-weight-column vectors, each with length r n k . Here weight of a vector means the number of 1 s in a vector. (a) Construct the parity-check matrix H of code C for n 16 and k 11. (b) Express the maximum code length of code C by using r. (c) Prove that sum of arbitrary two column vectors in H is an even-weight column vector. (d) Prove that the code C with maximum code length has the minimum Hamming distance 4. (e) Prove that every codeword of C has even weight.

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2.19 2.20

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If an error pattern e x is detectable for a cyclic code, show that its cyclic shifted pattern of e x is also detectable. Let g x be a generator polynomial with degree n k of the n; k cyclic code C. Prove that g x has the following properties: 1. g x is a monic polynomial with constant term equal to 1, and is uniquely determined. 2. g x is a factor of xn 1, meaning xn 1 g x h x , where h x , called a parity-check polynomial, has degree k. 3. The reciprocal of h x , de ned as xk h 1=x , is also a factor of xn 1, and generates a cyclic code, that is, a dual code of C.

2.21 2.22 2.23

For the 7; 4 cyclic code C de ned by the generator polynomial g x x3 x 1, show that the dual code of C has minimum Hamming distance 4. Given a 7; 4 cyclic SEC code C with a generator polynomial g x x3 x 1, obtain the dual code of C which has an SEC-DED capability. Prove Theorem 2.5.