### Table 3. Example: H Matrix for Parity-Check Code

"... In PAGE 20: ... If too few output codewords occur during normal operation to satisfy the self-testing requirement of the checkers, modifications such as those suggested in [Fujiwara 87] can be used. An example of a parity-check code is given in Table3 , and the block diagram for a self-checking circuit using this code is shown in Fig.... ..."

### Table 2: M-Code Parity Check Matrix for 32-bit Words

"... In PAGE 4: ... For example, for a 32-bit word, check bits are generated by XORing at most 14 bits, while Hamming code as used in[1] requires XORing up to 32 bits. Table2 represents the parity check matrix for a 32-bit word. Each row contains the bit positions to be XORed to generate each check bit.... ..."

### Table II. Example of a parity check matrix 1 0 0 0 0 0 1 0 0 0 0 1 0

2003

### Table 4.1. (20,3,4) parity check matrix 11110000000000000000 00001111000000000000 00000000111100000000 00000000000011110000 00000000000000001111 1 0001 0001 0001 0000000

2002

### Table V lists the number of rows in a redundant parity-check matrix needed to achieve a given stopping distance, for matrices constructed by the generalized HT method, the upper bound from [10], and a bound obtained from cyclic parity-check matrices. In addition, the upper bound from [2], given in Equation (10), is also shown. All results pertain to the [127, 113, 5] BCH code.

708

### Table III list the number of rows required to avoid elementary trapping sets of a given size, computed according to LLL and its high-probability variation. The values for m are derived using the minimum distance and the rank results taken from [22]. Observe that Table III indicates that there exists a parity-check matrix for the PG(2, 16) code with at most 173 rows and no (3, s), s lt; 45, elementary trapping sets. This is significantly less than n = 273. On the other hand, the LLL bound states that at most 516 rows are required to eliminate (16, s), s lt; 288 trapping sets, a significantly larger number than n.

2008

### Tables VI-IX contain (combined) parity check matrices of new codes. For 114 6149, the parameters printed are (length,dimension,constraint length,free distance). For 1146249, the parameters are (length,dimension,(constraint length vector),free distance), and the combined parity check matrices are printed in a compressed format. The matrices can be expanded as in the following example: Example. Let 114 6151. A combined parity check matrix for a 405359 5059 404959 4959 484159 5441 code is listed in compressed format as (6,20,23,24,37). Write these octal numbers as binary column vectors, least significant bit on top, and insert boundaries as prescribed by the constraint length vector. Thus

### Tables VI-IX contain (combined) parity check matrices of new codes. For 114 61 49, the parameters printed are (length,dimension,constraint length,free distance). For 114 62 49, the parameters are (length,dimension,(constraint length vector),free distance), and the combined parity check matrices are printed in a compressed format. The matrices can be expanded as in the following example: Example. Let 114 61 51. A combined parity check matrix for a 405359 5059 404959 4959 484159 5441 code is listed in compressed format as (6,20,23,24,37). Write these octal numbers as binary column vectors, least significant bit on top, and insert boundaries as prescribed by the constraint length vector. Thus

### Table 1: Description of the structure of the parity check matrices H(i); 1 i n in the example

2005

"... In PAGE 13: ...Table 1: Description of the structure of the parity check matrices H(i); 1 i n in the example the vector y = [ 1100011010010111000001010001111011111100 0101100010011011111111100000000100001111 0100110111001010010101011100111000001000 1001010011001110000110000001100110111111 00001101111011001101000101100111 ]T Coordinates of all codewords satisfy also the following family of parity check relations xi xi+2 xi+8 xi+12 xi+26 = 0; 1 i 166: One of the possible set of parity check matrices H(i); 1 i n = 192 can be described by the Table1 (the similar construction was used in [4]). For every i; 1 i n; the matrix H(i) is formed from that parity check relations (second column of the Table 1) for which the respective integer segment (with boundaries given in third and fourth columns) contains i.... In PAGE 13: ...Table 1: Description of the structure of the parity check matrices H(i); 1 i n in the example the vector y = [ 1100011010010111000001010001111011111100 0101100010011011111111100000000100001111 0100110111001010010101011100111000001000 1001010011001110000110000001100110111111 00001101111011001101000101100111 ]T Coordinates of all codewords satisfy also the following family of parity check relations xi xi+2 xi+8 xi+12 xi+26 = 0; 1 i 166: One of the possible set of parity check matrices H(i); 1 i n = 192 can be described by the Table 1 (the similar construction was used in [4]). For every i; 1 i n; the matrix H(i) is formed from that parity check relations (second column of the Table1 ) for which the respective integer segment (with boundaries given in third and fourth columns) contains i. In the Table 2 there were listed parameters connected with the matrices H(i): the number of rows ri, the number of non{zero columns ni, the number of non{zero columns after the application of Theorem 1, n0 i, and the numbers ki = ni ri, k0 i = n0 i ri, 1 i n.... ..."