2070 Asar (CS II) · 2 marks

Define Hamming weight and Hamming distance

Short answer

Hamming weight is the number of non-zero bits in a code vector. Hamming distance between two equal-length code vectors is the number of bit positions where they differ, equal to the weight of their bitwise XOR.

Define Hamming Weight and Hamming Distance.

Hamming weight $w(\mathbf{c})$ : the number of non-zero (1) bits in a code vector. Example: $w(1011001)=4$ .
Hamming distance $d(\mathbf{c}_1,\mathbf{c}_2)$ : the number of bit positions in which two code vectors of the same length differ; $d(\mathbf{c}_1,\mathbf{c}_2)=w(\mathbf{c}_1\oplus\mathbf{c}_2)$ . Example: $\mathbf{c}_1=1011001$ , $\mathbf{c}_2=1001011$ ; $\mathbf{c}_1\oplus\mathbf{c}_2=0010010$ , so $d=2$ .

The smallest distance between any pair of codewords, $d_{min}$ , sets the error capability: a code detects $d_{min}-1$ errors and corrects $\lfloor(d_{min}-1)/2\rfloor$ errors.

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