A First Course in Coding Theory by R. A. Hill

Machine Theory

By R. A. Hill

The purpose of this e-book is to supply an straightforward therapy of the idea of error-correcting codes, assuming not more than highschool arithmetic and the power to hold out matrix mathematics. The publication is meant to function a self-contained direction for moment or 3rd yr arithmetic undergraduates, or as a readable creation to the mathematical elements of coding for college kids in engineering or laptop technology.

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In this case the index variable is i whose type is first designated as int before setting it to 15. , i >= 0) that terminates the loop when it evaluates to false. Finally, direction is provided on how the loop should move through the values of the index variable. This is accomplished here with the i-- syntax which says that i should be decremented by one at each step through the loop. 4 provides more details on the use of loops in C++. The printBinary function picks off the binary representation of the unsigned int variable supplied for its argument by first shifting the integer 1 (= 20 ) over i = 15 slots using the 1 << 15 operation.

0000 × 2−2 . 00001 × 23 . 0001 if we round up. 5 under truncation and −12 if the answer is rounded up. This is an illustration of the types of problems that can occur when performing basic arithmetic operations using numbers that have very different magnitudes. 5 leaves it unchanged for any |a| ≤ 2−3 . Finally, let us see what happens when we try to store 1/5 using our simple system. This fraction cannot be stored exactly and, instead, the best that can be obtained is an approximate representation based on the fact that 1 5 .

This leads to S˜n = x ˜n (1 + En ) + S˜n−1 (1 + En ) = x ˜n (1 + En ) + x ˜n−1 (1 + En−1 )(1 + En ) + S˜n−2 (1 + En−1 )(1 + En ) n � ¯j ), = x ˜j (1 + E j=1 ¯ n = En , where E ¯j = 1+E n � (1 + Ek ) k=j ¯1 := E ¯2 . (The notation := that appears here and elsewhere throughfor j = 2, . . ) Hence, Sn − S˜n = j=1 x ˜j Ej and, as before ¯j | ≤ −1 + exp{(n − j)2−(m+1) }. |E But, unlike multiplication, the values of the x ˜j are inextricably linked into the relative approximation errors with the consequence that the bound depends on the order of summation.

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