BASIC CHARACTERISTICS OF BLOCKCODES

1 BASIC CHARACTERISTICS OF BLOCKCODES

Suppose C_i and C_j are any two code words in an (n, k) block code. A measure of the difference between the code words is the number of correspond­ing elements or positions in which they differ. This measure is called the Hamming distance between the two code words and is denoted as d_ij. Clearly, d_ij for i  ≠ j satisfies the condition 0 ≤ d_ij ≤ n. The smallest value of the set {d_ij} for the M code words is called the minimum distance of the code and is denoted as d _min. Since the Hamming distance is a measure of the separation between pairs of code words, it is intimately related to the cross-correlation coefficient between corresponding pairs of waveforms generated from the code words.

Besides characterizing a code as being binary or nonbinary, one can also describe it as either linear or non-linear. Suppose C_i and C_j are two code

words in an (n, k) block code and let α₁ and α₂ be any two elements selected from the alphabet. Then the code is said to be linear if and only if α₁ C_i + α₂ C_j is also a code word. This definition implies that a linear code must contain the all-zero code word.