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Binary Codes
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Binary codes are codes which are represented in binary system with modification from the original ones. Below we will be seeing the following: |
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- Weighted Binary Systems
- Non Weighted Codes
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Weighted Binary Systems
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Weighted binary codes are those which obey the positional weighting principles, each position of the number represents a specific weight. The binary counting sequence is an example. |
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Decimal
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8421
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2421
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5211
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Excess-3
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0
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0000
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0000
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0000
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0011
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1
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0001
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0001
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0001
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0100
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2
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0010
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0010
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0011
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0101
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3
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0011
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0011
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0101
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0110
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4
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0100
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0100
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0111
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0111
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5
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0101
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1011
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1000
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1000
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6
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0110
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1100
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1010
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1001
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7
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0111
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1101
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1100
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1010
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8
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1000
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1110
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1110
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1011
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9
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1001
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1111
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1111
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1100
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8421 Code/BCD Code
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The BCD (Binary Coded Decimal) is a straight assignment of the binary equivalent. It is possible to assign weights to the binary bits according to their positions. The weights in the BCD code are 8,4,2,1. |
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Example: The bit assignment 1001, can be seen by its weights to represent the decimal 9 because: |
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1x8+0x4+0x2+1x1 = 9 |
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2421 Code
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This is a weighted code, its weights are 2, 4, 2 and 1. A decimal number is represented in 4-bit form and the total four bits weight is 2 + 4 + 2 + 1 = 9. Hence the 2421 code represents the decimal numbers from 0 to 9. |
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5211 Code
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This is a weighted code, its weights are 5, 2, 1 and 1. A decimal number is represented in 4-bit form and the total four bits weight is 5 + 2 + 1 + 1 = 9. Hence the 5211 code represents the decimal numbers from 0 to 9. |
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Reflective Code
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A code is said to be reflective when code for 9 is complement for the code for 0, and so is for 8 and 1 codes, 7 and 2, 6 and 3, 5 and 4. Codes 2421, 5211, and excess-3 are reflective, whereas the 8421 code is not. |
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Sequential Codes
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A code is said to be sequential when two subsequent codes, seen as numbers in binary representation, differ by one. This greatly aids mathematical manipulation of data. The 8421 and Excess-3 codes are sequential, whereas the 2421 and 5211 codes are not. |
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Non Weighted Codes
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Non weighted codes are codes that are not positionally weighted. That is, each position within the binary number is not assigned a fixed value. |
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Excess-3 Code
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Excess-3 is a non weighted code used to express decimal numbers. The code derives its name from the fact that each binary code is the corresponding 8421 code plus 0011(3). |
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Example: 1000 of 8421 = 1011 in Excess-3 |
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Gray Code
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The gray code belongs to a class of codes called minimum change codes, in which only one bit in the code changes when moving from one code to the next. The Gray code is non-weighted code, as the position of bit does not contain any weight. The gray code is a reflective digital code which has the special property that any two subsequent numbers codes differ by only one bit. This is also called a unit-distance code. In digital Gray code has got a special place. |
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Decimal Number
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Binary Code
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Gray Code
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0
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0000
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0000
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1
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0001
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0001
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2
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0010
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0011
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3
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0011
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0010
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4
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0100
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0110
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5
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0101
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0111
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6
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0110
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0101
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7
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0111
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0100
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8
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1000
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1100
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9
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1001
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1101
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10
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1010
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1111
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11
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1011
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1110
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12
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1100
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1010
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13
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1101
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1011
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14
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1110
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1001
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15
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1111
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1000
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Binary to Gray Conversion
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- Gray Code MSB is binary code MSB.
- Gray Code MSB-1 is the XOR of binary code MSB and MSB-1.
- MSB-2 bit of gray code is XOR of MSB-1 and MSB-2 bit of binary code.
- MSB-N bit of gray code is XOR of MSB-N-1 and MSB-N bit of binary code.
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