# How to recognise the number of errors that can be detected and corrected of a large set of codewords (k) each having a specified number of bits (n)?

I am struggling to find how can I know the number of errors that can be corrected and detected using (n=10) bit code with a (k=550) codewords. As far as I know, to calculate the number of errors to be corrected and/or detected someone must know the Hamming distance between 2 codewords. However, I do not have any relevant information to find out a solution to this question. I tried to calculate the efficiency of this code using the formula: (E = Number of original message bits / length of the codeword). The efficiency in my case is 1/5. But, I do not know how can I use this information to calculate the number of errors that can be corrected and detected.

• Welcome to COMPUTER SCIENCE @SE. Can you please spell out how to arrive at E = 1/5? The numbers don't seem to fit. Dec 2, 2020 at 8:49
• E= 2/10 =1/5; since the number of original message bits is 2(binary can be 0 or 1). Dec 2, 2020 at 10:18
• You misunderstood what is meant by original message bits. This means the main terms that can be used to construct a set of codewords can be either 0 or 1. For example, 0010110010, 1100101100, 0101001110 are considered to be 3 codewords of 10 bits with a 2 as an original message bits. Dec 2, 2020 at 10:33
• the number of original message bits is 2 that sounds extremely unlikely - there would be exactly four messages possible. Can you quote or at least hyperlink the original problem? Dec 2, 2020 at 20:57

There are $$2^{10}$$ binary strings of length $$10$$. We can partition them into pairs $$x_1\ldots x_90,x_1\ldots x_91$$. Since your code contains more than $$2^9$$ codewords, it will contain at least one such pair. Hence the minimum distance of your code is only $$1$$.
More specifically, the set $$\{0,1\}^{10}$$ of all 1024 10-bit strings can be partitioned into 512 subsets of the form $$\{a,b\}$$, where $$a,b \in \{0,1\}^n$$, $$a$$ contains a 0 in the first position and $$b$$ contains a 1 in the first position, and $$a$$ and $$b$$ have the same values in the other positions. Thus, the Hamming distance between two codewords in the same subset is exactly $$1$$.