We will show that each of the three relations (3.11),(3.13),(3.14) between n, k, and d derived in Section4.2 implies (3.9), providing d is an integer in (3.11),(3.13) and d is an odd integer in (3.14). In order to make the appendix self-contained, we now re-state these inequalities:
|d 2 - 2||(11)|
|d 2 - 1||(12)|
Notice that what we are trying to establish has nothing to do with graphs or codes; this is just manipulation of integer inequalities. In particular, we have following simple lemma.
Lemma11. If a b/c and a, b, c are positive integers, then a (b + a)/(c + 1).
The proof of Lemma11 is straightforward, and is left to the reader. We first deal with (3.14), assuming d is odd. Taking the common denominator and applying (twice) Lemma11, we see that (3.14) implies
Since (d + 1)/2 is an integer for odd d, it follows from (20) that . This may be re-written as:
If n + 1 0 mod (k + 1) then , and (21) clearly implies (3.9). If (n + 1)/(k + 1) is an integer, then (21) is precisely the equivalent form of (3.9) given in (3.12).
It is easy to see that if d is an odd integer, then (3.11) implies (3.14). Since this case was already established above, it remains to prove (3.11) for even d. Using once again Lemma11, we see that (3.11) implies d2n/(k + 1), or equivalently d /2n/(k + 1). Since d /2 is an integer for even d, we can take the integer part of n/(k + 1) in the above expression. It follows that for even d, we have