MATH 245 CH11

# MATH 245 CH11 - DISCRETE MATHEMATICS W W L CHEN c W W L...

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Unformatted text preview: DISCRETE MATHEMATICS W W L CHEN c W W L Chen, 1991, 2008. This chapter is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners. Chapter 11 MATRIX CODES AND POLYNOMIAL CODES 11.1. Introduction In this section, we investigate how elementary group theory enables us to study coding theory more easily. Throughout this section, we assume that m,n ∈ N with n > m . Definition. Suppose that α : Z m 2 → Z n 2 is an encoding function. Then we say that C = α ( Z m 2 ) is a group code if C forms a group under coordinate-wise addition modulo 2 in Z n 2 . We denote by 0 the identity element of the group Z n 2 . Clearly 0 is the string of n 0’s in Z n 2 . PROPOSITION 11A. Suppose that α : Z m 2 → Z n 2 is an encoding function, and that C = α ( Z m 2 ) is a group code. Then min { δ ( x,y ) : x,y ∈ C and x 6 = y } = min { ω ( x ) : x ∈ C and x 6 = 0 } ; in other words, the minimum distance between strings in C is equal to the minimum weight of non-zero strings in C . Proof. Suppose that a,b,c ∈ C satisfy δ ( a,b ) = min { δ ( x,y ) : x,y ∈ C and x 6 = y } and ω ( c ) = min { ω ( x ) : x ∈ C and x 6 = 0 } . We shall prove that δ ( a,b ) = ω ( c ) by showing that (a) δ ( a,b ) ≤ ω ( c ); and (b) δ ( a,b ) ≥ ω ( c ). (a) Since C is a group, the identity element 0 ∈ C . It follows that ω ( c ) = δ ( c, 0) ∈ { δ ( x,y ) : x,y ∈ C and x 6 = y } , so ω ( c ) ≥ δ ( a,b ). Chapter 11 : Matrix Codes and Polynomial Codes page 1 of 12 Discrete Mathematics c W W L Chen, 1991, 2008 (b) Note that δ ( a,b ) = ω ( a + b ), and that a + b ∈ C since C is a group and a,b ∈ C . Hence δ ( a,b ) = ω ( a + b ) ∈ { ω ( x ) : x ∈ C and x 6 = 0 } , so δ ( a,b ) ≥ ω ( c ). Note that in view of Proposition 11A, we need at most ( |C|- 1) calculations in order to calculate the minimum distance between code words in C , compared to ( |C| 2 ) calculations. It is therefore clearly of benefit to ensure that C is a group. This can be achieved with relative ease if we recall Proposition 9E which we restate below in a slightly different form. PROPOSITION 11B. Suppose that α : Z m 2 → Z n 2 is an encoding function. Then the code C = α ( Z m 2 ) is a group code if α : Z m 2 → Z n 2 is a group homomorphism. 11.2. Matrix Codes – An Example Consider an encoding function α : Z 3 2 → Z 6 2 , given for each string w ∈ Z 3 2 by α ( w ) = w G , where w is considered as a row vector and where G = 1 0 0 1 1 0 0 1 0 0 1 1 0 0 1 1 0 1 . (1) Since Z 3 2 = { 000 , 001 , 010 , 011 , 100 , 101 , 110 , 111 } , it follows that C = α ( Z 3 2 ) = { 000000 , 001101 , 010011 , 011110 , 100110 , 101011...
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MATH 245 CH11 - DISCRETE MATHEMATICS W W L CHEN c W W L...

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