Confusing XOR notation in textbook sample solution

I am preparing for the midterm exam in my Data Security class so I am trying to read and understand the textbook's exercise questions. The sample solution at the end the textbook makes this problem more confusing instead of clarifying it. $\oplus$ denotes XOR operation but what are we trying to XOR with?

$c = \oplus c_i$ and $i \in I(c)$, but $c_i$ is getting XOR with what?

Any help would be appreciated.

Textbook question:

Textbook sample solution:

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I'm curious, what textbook is this? – d1str0 Feb 29 at 5:00
William Stallings, Cryptography and Network Security: Principles and Practice, 6/E. – Node.JS Feb 29 at 5:49
Thank you. I might want to check this out – d1str0 Feb 29 at 5:50

The notation $c=\oplus~c_i$ is (terrible) shorthand for $$c=\bigoplus_{i \in I(c)} c_i$$ where the sum sign should be replaced by the big xor sign which could also be written as $$c=\sum_{i \in I(c)} c_i,$$where $\sum$ denotes vector addition modulo 2.

An example of this decomposition (for length 8 vectors) is $$c=(1,0,1,0,0,0,1,0)=$$ which is nonzero in positions 1,3, and 7, and satisfies $$c=(1,0,0,0,0,0,0,0)\oplus(0,0,1,0,0,0,0,0)\oplus(0,0,0,0,0,0,1,0)$$ $$=c_1\oplus c_3 \oplus c_7=\bigoplus_{i \in I(c)} c_i$$ where $I(c)=\{1,3,7\}.$

Then, since $E$ is assumed to be linear, all the places you marked with an arrow make sense.

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Awesome! Thank you so much for clarification. – Node.JS Feb 29 at 4:42
+1 for (terrible) shorthand :D – user32017 Feb 29 at 17:57