# Understanding connection between length of codeword and hamming distance in Hamming code

I came across following in Huffman coding:

Minimum Hamming distance to correct up to s errors is $$2s + 1$$ because that way the legal codewords are so far apart that even with $$s$$ changes the original codeword is still closer than any other codeword.

Then I came across following:

Consider dataword length of $$m$$ bits, codeword length of $$n$$ bits. So redundant check bits will be $$n-m = r$$. Each of the $$2^m$$ legal messages has $$n$$ illegal codewords at a distance of $$1$$ from it. These are formed by systematically inverting each of the $$n$$ bits in the $$n$$-bit codeword formed from it. Thus, each of the $$2^m$$ legal messages requires $$n+1$$ bit patterns dedicated to it. Since the total number of bit patterns is $$2^n$$, we must have $$(n+1) 2^m ≤ 2^n$$.

Using this information, I took $$m=4$$ and found that to meet above equation, $$n \ge 7$$. So with $$n = 7$$, I prepared Huffman code for $$m = 4$$ as shown in the image ($$d$$ are data bits, $$c$$ are check bits): The code has hamming distance of 3 which confirms with $$2(1)+1 = 3$$ as required to correct 1 bit errors.

Q1. But then, how satisfying $$(n+1) 2^m ≤ 2^n$$ also ensures hamming distance is 3?

Q2. Also what will be equation if I want to correct $$k$$ bit errors? Will it be $$(\binom{n}{k}+1)2^m\leq 2^n$$?

Consider a binary code on $$n$$ bits with minimum distance $$2d+1$$ and $$M$$ codewords. Imagine surrounding each point of the code with a ball of radius $$d$$, consisting of all points at distance at most $$d$$ from the point. These balls must be disjoint: if the balls corresponding to $$x,y$$ contain some point $$z$$ in common (note that $$z$$ need not be in the code), then $$d(x,y) \leq d(x,z) + d(y,z) \leq 2d$$, contradicting the stated minimum distance. Each ball contains $$\binom{n}{\leq d} := \sum_{r=0}^{d} \binom{n}{r}$$ points, and so $$M \times \binom{n}{\leq d} \leq 2^n.$$ This is because there are $$M$$ disjoint balls, each containing $$\binom{n}{\leq d}$$ points, while in total there are $$2^n$$ binary strings of length $$n$$.
• "Surrounding each point of the code with a ball of radius $d$, consisting of all points at distance at most $d$ from the point". Doesnt that mean each ball contains $\binom{n}{\leq d}$ points (given a codeword corrsponding to the center of the ball, we choose any $r (\leq d)$ bit positions out of $n$ to get illegal codewords at distance at most $d$ from the center)? What I am missing? – anir Mar 10 at 19:49
• Bit confused. Does above answer resolve my Q2? Does it mean, to correct $k$ errors, we need $d=2k+1$ & then follow $M\times \binom{n}{\leq d}\leq 2^n$? For correcting $k$ bit errors, we have $d=2k+1$ & need $M\times \binom{n}{\leq(2k+1)}\leq 2^n$. Q3. Is it so? Q4. Also are two inequalities comparable: $(n+1)2^m≤2^n$ and $M×\binom{n}{≤d}≤2^n$. That is, should $\binom{n}{≤d}==n+1$>? **Q5.** I [read](en.wikipedia.org/wiki/Hamming_code) that $d=3$ for Hamming code & can correct only 1 bit errors. Cant we design Hamming code with $d>3$ to correct more than 1 bit errors? – anir Mar 11 at 19:42
• Your $k$ and my $d$ are the same. Sorry for the confusion. – Yuval Filmus Mar 11 at 19:53