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In my textbook, the encoding matrix H was given as the matrix such that

if U is a codeword then HU

^{T}is the zero vector.

In this case, the number of columns in H would be the length of the

**codeword**.

But in another explanation, I read that an encoding matrix H is such that if M is the message and U the codeword with the check digits appended, HM = U.

In this case, the number of columns would be the length of the

**message**.

Please help me clarify.

This question arose in the context of the following question, so I may ask questions related to it later, after I understand better what exactly an encoding matrix is.

To transmit the eight possible messages of three binary symbols, a code appends three further symbols: if the message is ABC, the code transmits ABC(A+B)(A+C)(B+C). Write down the encoding matrix for this code. How many errors can it detect and correct?