Error Correcting Codes - Hamming codes

Errors and Error Correcting Codes

When bits are transmitted over the computer network, they are subject to get corrupted due to interference and network problems. The corrupted bits leads to spurious data being received by the receiver and are called errors.

Error-correcting codes (ECC) are a sequence of numbers generated by specific algorithms for detecting and removing errors in data that has been transmitted over noisy channels. Error correcting codes ascertain the exact number of bits that has been corrupted and the location of the corrupted bits, within the limitations in algorithm.

ECCs can be broadly categorized into two types −

• Block codes − The message is divided into fixed-sized blocks of bits, to which redundant bits are added for error detection or correction.

• Convolutional codes − The message comprises of data streams of arbitrary length and parity symbols are generated by the sliding application of a Boolean function to the data stream.

Hamming Code

Hamming code is a block code that is capable of detecting up to two simultaneous bit errors and correcting single-bit errors. It was developed by R.W. Hamming for error correction.

In this coding method, the source encodes the message by inserting redundant bits within the message. These redundant bits are extra bits that are generated and inserted at specific positions in the message itself to enable error detection and correction. When the destination receives this message, it performs recalculations to detect errors and find the bit position that has error.

Encoding a message by Hamming Code

The procedure used by the sender to encode the message encompasses the following steps −

• Step 1 − Calculation of the number of redundant bits.

• Step 2 − Positioning the redundant bits.

• Step 3 − Calculating the values of each redundant bit.

Once the redundant bits are embedded within the message, this is sent to the user.

Step 1 − Calculation of the number of redundant bits.

If the message contains m𝑚number of data bits, r𝑟number of redundant bits are added to it so that m𝑟 is able to indicate at least (m + r+ 1) different states. Here, (m + r) indicates location of an error in each of (𝑚 + 𝑟) bit positions and one additional state indicates no error. Since, r𝑟 bits can indicate 2^r𝑟 states, 2^r𝑟 must be at least equal to (m + r + 1). Thus the following equation should hold  2^r ≥ m+r+1

Step 2 − Positioning the redundant bits.

The r redundant bits placed at bit positions of powers of 2, i.e. 1, 2, 4, 8, 16 etc. They are referred in the rest of this text as r₁ (at position 1), r₂ (at position 2), r₃ (at position 4), r₄ (at position 8) and so on.

Step 3 − Calculating the values of each redundant bit.

The redundant bits are parity bits. A parity bit is an extra bit that makes the number of 1s either even or odd. The two types of parity are −

• Even Parity − Here the total number of bits in the message is made even.

• Odd Parity − Here the total number of bits in the message is made odd.

Each redundant bit, r_i, is calculated as the parity, generally even parity, based upon its bit position. It covers all bit positions whose binary representation includes a 1 in the i^th position except the position of r_i. Thus −

• r₁ is the parity bit for all data bits in positions whose binary representation includes a 1 in the least significant position excluding 1 (3, 5, 7, 9, 11 and so on)

• r₂ is the parity bit for all data bits in positions whose binary representation includes a 1 in the position 2 from right except 2 (3, 6, 7, 10, 11 and so on)

• r₃ is the parity bit for all data bits in positions whose binary representation includes a 1 in the position 3 from right except 4 (5-7, 12-15, 20-23 and so on)

Decoding a message in Hamming Code

Once the receiver gets an incoming message, it performs recalculations to detect errors and correct them. The steps for recalculation are −

• Step 1 − Calculation of the number of redundant bits.

• Step 2 − Positioning the redundant bits.

• Step 3 − Parity checking.

• Step 4 − Error detection and correction

Step 1 − Calculation of the number of redundant bits

Using the same formula as in encoding, the number of redundant bits are ascertained.

2^r ≥ m + r + 1 where m is the number of data bits and r is the number of redundant bits.

Step 2 − Positioning the redundant bits

The r redundant bits placed at bit positions of powers of 2, i.e. 1, 2, 4, 8, 16 etc.

Step 3 − Parity checking

Parity bits are calculated based upon the data bits and the redundant bits using the same rule as during generation of c₁,c₂ ,c₃ ,c₄ etc. Thus

c₁ = parity(1, 3, 5, 7, 9, 11 and so on)

c₂ = parity(2, 3, 6, 7, 10, 11 and so on)

c₃ = parity(4-7, 12-15, 20-23 and so on)

Step 4 − Error detection and correction

The decimal equivalent of the parity bits binary values is calculated. If it is 0, there is no error. Otherwise, the decimal value gives the bit position which has error. For example, if c₁c₂c₃c₄ = 1001, it implies that the data bit at position 9, decimal equivalent of 1001, has error. The bit is flipped to get the correct message.