Elias Omega Coding

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 F₁ = |β(n)|, using efficient and distinguishable encoding, then she can send the actual beta code F₂ = 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 ⌊log₂n⌋ bits of 0 (⌊x⌋ is the floor of x). The second field, the postfix, is the actual binary representation of n using ⌊log₂n⌋ + 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 (⌊log₂n⌋) of the gamma code and Elias omega code applies a recursion over the prefix Elias gamma representation of ⌊log₂n⌋.

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 requires 29 bits. Hence, in the first step, it is encoded as follows:

〈F₁, F₂〉 = 〈0000000000000000000000000000, 11111111111111111111111111011〉

where blanks, 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:

1. 〈28 0's, β(536870907)〉 =
   〈0000 0000 0000 0000 0000 0000 0000, 1 1111 1111 1111 1111 1111 1111 1011〉
2. 〈〈4 0's, β(28)〉, β(536870907)〉 =
   〈〈0000, 1 1100〉, 1 1111 1111 1111 1111 1111 1111 1011〉
3. 〈〈〈2 0's, β(4)〉, β(28)〉, β(536870907)〉 =
   〈〈〈00, 100〉, 1 1100〉, 1 1111 1111 1111 1111 1111 1111 1011〉
4. 〈〈〈〈one 0, β(2)〉, β(4)〉, β(28)〉, β(536870907)〉 =
   〈〈〈〈0, 10〉, 100〉, 1 1100〉, 1 1111 1111 1111 1111 1111 1111 1011〉

Taking away all the blanks, 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.

Problem statement

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

Input Specification

The input contains positive integers, each in a separate line. Each integer is between 2 and 2,000,000,000, inclusive. The end of input is indicated by a zero. The input can contain up to one hundred lines.

Output Specification

The output consists of lines, each corresponding to an input integer except the last zero. Each line contains the Elias omega code of the respective input integer. The output should not contain any blanks or blank lines.

Sample Input

1. 2
2. 510
3. 7
4. 120000
5. 536870905
6. 49
7. 5
8. 0

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Output for Sample Input

1. 010
2. 0111000111111110
3. 010111
4. 0101001000011101010011000000
5. 0101001110011111111111111111111111111001
6. 010101110001
7. 010101

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