The Gray code is sometimes called a reflected code because the bit values are reflected on both sides of any 2″ value

The Gray code is sometimes called a reflected code because the bit values are reflected on both sides of any 2″ value. For example, as shown in Table 3-5, the values of the three low-order bits are reflected over a line drawn between 7 and 8. Using this property of the Gray code, obtain:

a. The Gray code numbers for 16 through 31 as a continuation of Table 3-5.

b. The excess-3 Gray code for decimals 10 to 19 as a continuation of the list in Table 3-6.

The Correct Answer and Explanation is :

Part (a): Gray Code Numbers for 16 through 31

To generate the Gray code numbers for 16 through 31, we start by recalling the Gray code conversion method. The Gray code for a number (N) is calculated by performing an XOR operation between (N) and the number (N >> 1) (right-shifted value of (N)).

The table of Gray codes from 16 to 31 is obtained by following this procedure.

Decimal	Binary	Gray Code
16	10000	11000
17	10001	11001
18	10010	11011
19	10011	11010
20	10100	11110
21	10101	11111
22	10110	11101
23	10111	11100
24	11000	10100
25	11001	10101
26	11010	10111
27	11011	10110
28	11100	10010
29	11101	10011
30	11110	10001
31	11111	10000

Explanation:

• The Gray code for any binary number can be generated by XORing the binary number with itself right-shifted by one bit. For example, for 16, in binary it’s (10000), and its Gray code is generated as follows: XOR(10000, 01000) = 11000.
• The table above demonstrates the conversion of decimal numbers 16 through 31 to Gray code using this method.

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Part (b): Excess-3 Gray Code for Decimals 10 to 19

Excess-3 is a binary-coded decimal (BCD) system where each decimal digit is represented by its corresponding binary equivalent plus 3. After this transformation, the result is used in generating the Gray code.

To get the Excess-3 Gray code, first add 3 to each decimal number (10 to 19) and convert the result to binary. Then, convert this binary value to Gray code.

Decimal	Excess-3	Binary (Excess-3)	Gray Code (Excess-3)
10	13	1101	1111
11	14	1110	1010
12	15	1111	1011
13	16	10000	11000
14	17	10001	11001
15	18	10010	11011
16	19	10011	11010
17	20	10100	11110
18	21	10101	11111
19	22	10110	11101

Explanation:

• Excess-3 encoding: First, add 3 to the decimal values (10 to 19), which results in the numbers 13 to 22. These are then converted into binary, and the Gray code is generated as described in part (a).
• For example, for decimal 10, Excess-3 gives 13, which in binary is (1101), and its Gray code is (1111).

Thus, the two tables display the Gray code transformations for the specified ranges in the question.