LINEAR BLOCK CODER USING ROM

LINEAR BLOCK CODER USING ROM

When the codeword corresponding to each uncoded word has been deter­mined, the results can be stored in a memory. Such a memory, in this case

a read­ only memory (ROM), is shown in Fig. 2.2.The bits x_ml  x_m2 . . . x_mk of the uncoded word are presented as an address to the memory. At the memory location speci­fied by that address we have stored the n bits corresponding to the coded word. The uncoded word bit serial bit stream x_i is converted to a parallel bit array (i.e., all bits available at the same time) by a serial to parallel converter (i.e., a shift register). The converter is driven by a clock at the uncoded bit rate f_b. When the bits x_ml  x_m2 . . . x_mk are all available, a READ command is given to the ROM and a LOAD command to the parallel to series converter (i.e., another shift register with a parallel load facility). The coded word c_m1 c_m2 … c_mn is thus loaded into the converter. The converter then presents the bits serially at its output. The clock rate for this second converter is f_c which is

the bit rate for the coded bits, where f_c/f_b = n / k

                      

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.