6.26 Error-correcting codes: hamming distance

So-called linear codes create error-correction bits by combining the data bits linearly. Topics discussed include generator matrices and the Hamming distance.

So-called linear codes create error-correction bits by combining the data bits linearly. The phrase "linearcombination" means here single-bit binary arithmetic.

 $0\mathop{\mathrm{xor}}0=0$ $1\mathop{\mathrm{xor}}1=0$ $0\mathop{\mathrm{xor}}1=1$ $1\mathop{\mathrm{xor}}0=1$ $·(0, 0)=0$ $·(1, 1)=1$ $·(0, 1)=0$ $·(1, 0)=0$
For example, let's consider the specific (3, 1) error correction code described by the following coding table and,more concisely, by the succeeding matrix expression. $\begin{array}{c}c(1)=b(1)\\ c(2)=b(1)\\ c(3)=b(1)\end{array}$ or $\begin{array}{c}c=Gb\end{array}$ where $\begin{array}{c}G=\begin{pmatrix}1\\ 1\\ 1\\ \end{pmatrix}\\ c=\left(\begin{array}{c}c(1)\\ c(2)\\ c(3)\end{array}\right)\\ b=\left(\begin{array}{c}b(1)\end{array}\right)\end{array}$

The length- $K$ (in this simple example $K=1$ ) block of data bits is represented by the vector $b$ , and the length- $N$ output block of the channel coder, known as a codeword , by $c$ . The generator matrix $G$ defines all block-oriented linear channel coders.

As we consider other block codes, the simple idea of the decoder taking a majority vote of the received bitswon't generalize easily. We need a broader view that takes intoaccount the distance between codewords. A length- $N$ codeword means that the receiver must decide among the $2^{N}$ possible datawords to select which of the $2^{K}$ codewords was actually transmitted. As shown in [link] , we can think of the datawords geometrically. We define the Hamming distance between binary datawords ${c}_{1}$ and ${c}_{2}$ , denoted by $d({c}_{1}, {c}_{2})$ to be the minimum number of bits that must be "flipped" to gofrom one word to the other. For example, the distance between codewords is 3 bits. In our table of binary arithmetic, wesee that adding a 1 corresponds to flipping a bit. Furthermore, subtraction and addition are equivalent. We can express theHamming distance as

$d({c}_{1}, {c}_{2})=\mathrm{sum}({c}_{1}\mathop{\mathrm{xor}}{c}_{2})$

Show that adding the error vector col[1,0,...,0]to a codeword flips the codeword's leading bit and leaves the rest unaffected.

In binary arithmetic (see [link] ), adding 0 to a binary value results in that binary value while adding 1results in the opposite binary value.

The probability of one bit being flipped anywhere in a codeword is $N{p}_{e}(1-{p}_{e})^{(N-1)}$ . The number of errors the channel introduces equals the number of ones in $e$ ; the probability of any particular error vector decreases with the number of errors.

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