I want to know which operations are valid for predicting/finding a new codeword from a set of known codewords for a linear block code having linear encoder mapping,

for eg.:
Consider a linear Hamming code (having linear encoder mapping) is used to map $4$ - bit messages to $7$ - bit codewords,
if it is known that the message $0001$ is mapped to the codeword $0000111$,
i.e, $$0001 \, \to \, 0000111 \, \dots (i)$$ $$\& \quad 0011 \, \to \, 1100110 \, \dots (ii)$$ ,then we can find out the corresponding codeword for the message $0010$ as follows:
$\because 0010 = (0001) \, \oplus \, (0011)$ , And we know EXOR is a valid operation $$\therefore \text{codeword for } 0010 \to (0000111) \, \oplus \, (1100110) $$ $$=1100001$$ Now, can we apply any other operations (like AND , OR , EXNOR , ...) to find the corresponding message codeword?

  • 1
    $\begingroup$ The codeword is the modulo-2 product of the dataword with a code generator matrix. This product is defined in terms of modulo-2 addition (XOR). Replacing the XOR with other operations does not make sense in this context. $\endgroup$ – MBaz Feb 24 '19 at 20:25

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