Binary codes are used to represent distinct discrete element of information. They are patterns of 1s and 0s for computer to understand information other than binary numbers.

The discrete elements of information is not only binary numbers, but also, other types of information such as decimal numbers, etc.

## Binary Code

Suppose we have a n-bit code then there are 2^{n} combination of binary codes consists of 1s and 0s.

For example,

3-bit code has 2^{3} = 8 codes

Bit Combination | Codes |
---|---|

0 | 000 |

1 | 001 |

2 | 010 |

3 | 011 |

4 | 100 |

5 | 101 |

6 | 110 |

7 | 111 |

There are 8 combinations, but the bit combination has value is between 0 to 2^{n} – 1. A 3-bit code has bit combination from 0 to 2^{3} – 1 = 7.

## Minimum bits required

For 16 code combinations, you need minimum of 4 bits, n = 4 bits, so that 2^{4 }= 16.

For 8 codes, you need minimum 3 bits like in example above,

n = 3 bits so that 2^{3 }= 8.

## Maximum bit for Binary Code

There is no restriction on number of bits that you can use for a binary code.

For example

10 decimal numbers – 0,1,2,3,4,5,6,7,8,9 can be represented using 4-bits and can also be represented using 10-bits. Let’s check this for first 5 decimal numbers.

Decimal value | 4-bit code | 10-bit code |
---|---|---|

0 | 0000 | 0000000000 |

1 | 0001 | 0000000010 |

2 | 0010 | 0000000100 |

3 | 0011 | 0000001000 |

4 | 0100 | 0000010000 |

A 4-bit code is binary representation of decimal and 10-bit code uses placeholder for each decimal number. Here the code – 0000010000 does not mean 2^{5} = 16, but it means 5^{th} position from right is a 1, therefore, decimal number = 4.

## Bibliography

John.F.Wakerly. 2008. *Digital Design: Principles And Practices, 4/E.* Pearson Education, India.

Mano, M. Morris. 1984. *Digital Design.* Pearson.

NATARAJAN, ANANDA. 2015. *Digital Design.* PHI Learning Pvt. Ltd.