Question

Which of these codes are prefix codes?

a) a: 11, e: 00, t: 10, s: 01

b) a: 0, e: 1, t: 01, s: 001

c) a: 101, e: 11, t: 001, s: 011, n: 010

d) a: 010, e: 11, t: 011, s: 1011, n: 1001, i: 10101

Short answer

1. Yes
2. No
3. Yes
4. Yes

Step-by-step solution

(a) Consider the code a:11, e:00, t:10, s:01.

The bit string 11 represents the letter ‘a’ and the letter ‘a’ not contains other letters e, t,s in the sequence of bits in it.

The bit string 00 represents the letter ‘e’ and the letter ‘e’ not contains other letters a, t,s in the sequence of bits in it.

The bit string 10 represents the letter ‘t’ and the letter ‘t’ not contains other letters a,e,s in the sequence of bits in it.

The bit string 01 represents the letter ‘s’ and the letter ‘s’ not contains other letters a,e,t in thesequence of bits in it.

Therefore, by the definition of prefix codes, the given code is a prefix code.

(b) Consider the code a:0,e:1,t:01,s:001.

The bit string 0 represents the letter ‘a’ and the letter ‘a’ not contains other letters e, t,s in the sequence of bits in it.

The bit string 1 represents the letter ‘e’ and the letter ‘e’ not contains other letters a, t,s in the sequence of bits in it.

The bit string 01 represents the letter ‘t’ and the letter ‘t’ is the sequence of other letters a and e.

The bit string 001 represents the letter ‘s’ and the letter ‘s’ is the sequence of other letters a and t.

Hence, by the definition of prefix codes, the given code is not a prefix code.

(c) Consider the code a: 101, e: 11, t: 001, s: 011, n: 010.

The bit string 101 represents the letter ‘a’ and the letter ‘a’ not a sequence of other letters e,t,s,n.

The bit string 11 represents the letter ‘e’ and the letter ‘e’ not a sequence of other letters a, t,s,n.

The bit string 001 represents the letter ‘t’ and the letter ‘t’ not a sequence of other letters a,e,s,n.

The bit string 011 represents the letter ‘s’ and the letter ‘s’ not contains other letters a,e, t,n in the sequence of bits in it.

The bit string 010 represents the letter ‘n’ and the letter ‘n’ not contains other letters a,e, t,s in the sequence of bits in it.

So, by the definition of prefix codes, the given code is a prefix code.

(d) Consider the code a: 010, e: 11, t: 011, s: 1011, n: 1001, i: 10101.

The bit string 010 represents the letter ‘a’ and the letter ‘a’ not a sequence of other letters e, t,s,n, i.

The bit string 11 represents the letter ‘e’ and the letter ‘e’ not a sequence of other letters a, t,s,n, i.

The bit string 011 represents the letter ‘t’ and the letter ‘t’ not a sequence of other letters a,e,s,n, i.

The bit string 1011 represents the letter ‘s’ and the letter ‘s’ not a sequence of other letters a,e, t,n, i.

The bit string 1001 represents the letter ‘n’ and the letter ‘n’ not contains other letters a,e, t,s,i in the sequence of bits in it.

The bit string 10101 represents the letter ‘i’ and the letter ‘i’ not contains other letters a,e,t,s,n in the sequence of bits in it.

Consequently, by the definition of prefix codes, the given code is a prefix code.