A first course in coding theory by Raymond Hill

By Raymond Hill

Algebraic coding conception is a brand new and quickly constructing topic, renowned for its many useful functions and for its fascinatingly wealthy mathematical constitution. This booklet offers an straight forward but rigorous creation to the idea of error-correcting codes. in accordance with classes given by means of the writer over numerous years to complex undergraduates and first-year graduated scholars, this consultant features a huge variety of workouts, all with ideas, making the booklet hugely compatible for person examine

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First observe that the round-to-closest value choice for x ˜ is obtained by increasing bm by one if bm+1 = 1 and using the chopped approximation otherwise. Consequently,   m � x ˜ = 2p  bj 2−j + ˜bm+1 2−(m+1)  j=0 with ˜bm+1 being either 2 or 0 depending on whether or not bm+1 is 1 or 0, respectively. 22 COMPUTER REPRESENTATION OF NUMBERS Thus, when bm+1 = 1 the error is � � � � ∞ � � � |x − x ˜| = 2p ��2−(m+1) − bj 2−j �� ≤ 2p−(m+1) . � � j=m+2 On the other hand, if bm+1 = 0, x− x ˜ ≤ 2p−(m+1) from our analysis of rounding by chopping.

By now writing n � (1 + Ei ) = 1 + E j=2 bounding the relative error |Pˆn − P˜n |/|P˜n | becomes tantamount to placing a bound on |E|. 5) we have � �n−1 � �n−1 −1 + 1 − 2−(m+1) ≤ E ≤ −1 + 1 + 2−(m+1) with � �n−1 −1 + 1 − 2−(m+1) = n−1 �� j=1 � n−1 (−2)−j(m+1) j � n − 1 −j(m+1) 2 j j=1 � � �n−1 � = − −1 + 1 + 2−(m+1) . ≥ − n−1 �� 24 COMPUTER REPRESENTATION OF NUMBERS Thus, � �n−1 |E| ≤ −1 + 1 + 2−(m+1) n−1 � �n − 1� = −1 + 2−j(m+1) j j=0 ≤ −1 + ∞ � (n2−(m+1) )j j=0 j! = −1 + exp{n2−(m+1) }. So, if n is small compared to 2m+1 , the relative error in approximating P˜n by Pˆn will be similarly small and on the order of n2−(m+1) .

2. cpp binaryRep . 2 employs some elementary bit-wise operations in a function called printBinary to pick off the internal binary representation of an integer. The first thing we see in the listing is a prototype or declaration of the basic form for printBinary that appears prior to the function main. , the keyword that immediately precedes the function’s name) is designated as void which means it does not return a value to the calling program. , return type and type for the arguments) in a function before it can be used.

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