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POGenerating all n-bit strings whose hamming distance is n/2
 

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  1. POGenerating all n-bit strings whose hamming distance is n/2
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    <p>I'm playing with some variant of <a href="http://en.wikipedia.org/wiki/Hadamard_matrix" rel="nofollow">Hadamard matrices</a>. I want to generate <strong>all</strong> <i>n</i>-bit binary strings which satisfy these requirements:</p> <ol> <li>You can assume that <i>n</i> is a multiple of 4. </li> <li>The first string is 0<sup>n</sup>.<br />&rarr; a string of all 0s.</li> <li>The remaining strings are sorted in alphabetic order.<br />&rarr; 0 comes before 1.</li> <li>Every two distinct <i>n</i>-bit strings have <a href="http://en.wikipedia.org/wiki/Hamming_distance" rel="nofollow">Hamming distance</a> <i>n/2</i>.<br />&rarr; Two distinct <i>n</i>-bit strings agree in exactly <i>n/2</i> positions and disagree in exactly <i>n/2</i> positions.</li> <li>Due to the above condition, every string except for the first string must have the same number of 0s and 1s.<br/> &rarr; Every string other than the first string must have <i>n/2</i> ones and <i>n/2</i> zeros.</li> <li>(<em>Updated</em>) All the <i>n</i>-bit strings begin with <code>0</code>.</li> </ol> <p>For example, this is the list that I want for when <i>n=4</i>.</p> <blockquote> <p><code>0000</code><br /> <code>0011</code><br /> <code>0101</code><br /> <code>0110</code></p> </blockquote> <p>You can easily see that every two distinct rows have hamming distance <i>n/2 = 4/2 = 2</i> and the list satisfies all the other requirements as well.</p> <p>Note that I want to generate <strong>all</strong> such strings. My algorithm may just output three strings <code>0000</code>, <code>0011</code>, and <code>0101</code> before terminating. This list satisfies all the requirements above but it misses <code>0110</code>.</p> <ol> <li>What would be a good way to generate such sets? <br />A python pseudo-code is preferred but any high-level description will do.</li> <li>What is the <strong>maximum</strong> number of such strings for a given <i>n</i>?<br />For example, when <i>n=4</i>, the max number of such strings happen to be 4. I'm wondering whether there can be any closed form solution for this upper bound.</li> </ol> <p>Thanks.</p>
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    1. USuser19906
    1. USuser19906
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      1. USגלעד ברקן
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    1. COThese lists are not unique. For example, here are 3 other lists that satisfy your criteria: {0000,0110,1010,1100}, {0000,0011,1001,1010}, {0000,0101,1001,1100}. Do you also want to generate all such lists?
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      1. USMatt
    2. CO@Matt Oops. I forgot to mention it. All the n-bit strings begin with 0. Then I think it guarantees that there exists only one such a list. Sounds reasonable?
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    3. COCould you post an example for n = 8 ?
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      1. USuser1952500
 

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