If the checkbox is not checked, it is not sent to the server. I suggest you to do something like this:
<label for="b0" class="left">Item 1</label>
<input type="hidden" name="b[0]" value=""/>
<input type="checkbox" name="b[0]" id="b0" value="Item 1"/>
<label for="b1" class="left">Item 2</label>
<input type="hidden" name="b[1]" value=""/>
<input type="checkbox" name="b[1]" id="b1" value="Item 2"/>
<label for="b2" class="left">Item 3</label>
<input type="hidden" name="b[2]" value=""/>
<input type="checkbox" name="b[2]" id="b2" value="Item 3"/>
This way, you are sure that b[0], b[1], etc. are always sent