Introduction


Elias code can be used to efficiently encode positive integers when there is no prior
information about the cardinality of the integers to be encoded and it is known that the
probability of getting a large integer is smaller than or equal to the probability of getting a small
integer. For practical reasons, however, in this contest we do pose a limit on the size of the
input integers. For the same reason we restrict the input integer to be greater than 1.

Elias developed three variants of the code: the Elias gamma, Elias delta, and Elias omega
coding methods. This problem presents the Elias gamma and Elias omega methods and calls for
implementing the Elias omega code. The following is a background, definition, and illustration
of the problem.

Background


Suppose that Alice wants to transmit a positive integer n to Bob through a binary channel

and let β(n) stand for the binary representation of n. If Bob knows |β(n) | (the number of bits
required for the binary representation of n) in advance, then Alice should use β(n) for the
transmission. On the other hand, if Bob does not have this information, then Alice can first send
|β(n)|, using efficient and distinguishable encoding, then she can send the actual beta code

F1= β(n). The end result is a two field code < F1,F2 > .
Elias code and its variants differ in the way they encode these two pieces of information
(F1 and F2). The main difference between variants lies in the representation of F1 . This may
imply modifications in the representation of F1 . In addition, some of the variants apply repetition
or recursion to the representation of F1. We use a specific variant specified below.

Definition


Formally, in Elias gamma coding, a positive integer n is encoded using two concatenated bit
fields. The first field, the prefix, contains [log₂ n] bits of 0 ( [x] is the floor of x). The second
field, the postfix, is the actual binary representation of n using [log₂ n]+1 bits. For example, the
binary representation of the decimal number 9 is 1001. Under Elias coding 9 is encoded as
0001001. The first three leading zeros denote that four bits are required for the binary
representation of 9. The next four bits contain the binary representation of 9. Elias delta code
applies the gamma code to the prefix([log₂ n]) of the gamma code and Elias omega code applies
a recursion over the prefix Elias gamma representation of [log₂ n].

Illustration (detailed example)


To further illustrate the Elias omega code, consider the integer 536870907. The binary
representation of this integer is 1 1111 1111 1111 1111 1111 1111 1011 which is required 29
bits. Hence, in the first step it is encoded as follows:

< F1,F2 > =
< 0000 0000 0000 0000 0000 0000 0000, 1 1111 1111 1111 1111 1111 1111 1011>

where blanks, dots, commas, and brackets, are inserted for readability.
To emphasize, for this contest the recursion stops when the first field of a recursive
stage contains one 0.

Looking at all the bits generated by all the steps and using β(n) to denote the binary
representation of n, we have:

a. <28 0’s, β(536870907)>  = 

 < 0000 0000 0000 0000 0000 0000 0000, 1 1111 1111 1111 1111 1111 1111 1011>

b. < <4 0’s, β(28)>, β(536870907) >    = 

< <0000, 1 1100>, 1 1111 1111 1111 1111 1111 1111 1011> 

c. <<<2 0’s, β(4)>, β(28)> , β(536870907) > = 

< <<00, 100>, 1 1100>, 1 1111 1111 1111 1111 1111 1111 1011> 

d. <<<, β(4)>, β(28)>, β(536870907 ) > = 

                        < <<<0, 10>, 100>, 1 1100>, 1 1111 1111 1111 1111 1111 1111 1011> 

Taking away all the blanks, dots, commas, and brackets we get that the Elias omega code
of 536870907 is: 0101001110011111111111111111111111111011. This code is distinguishable
(uniquely decodable). It can be decoded in a unique way and yield back the number 536870907.

Given some positive integers in the range of 2 to 2*10⁹
, you are to write a program to produce the
Elias omega codes for these integers.