# Linear Code

If the linear combination of codewords is also a codeword, the code is called a [linear code](https://en.wikipedia.org/wiki/Linear_code).

$$
c\_1, c\_2 \in C \land a \cdot c\_1 + b \cdot c\_2 \in C
$$

where $$a, b \in \mathbb{F}\_q$$.

**Reed-Solomon (RS) codes**, which are commonly used, satisfy this property and are therefore linear codes. (This property allows the folding operation to be used in RS-based FRI.)

$$
\mathsf{Enc}(f\_1) = {f\_1(\omega), f\_1(\omega^2), f\_1(\omega^3), f\_1(\omega^4)} \\
\mathsf{Enc}(f\_2) = {f\_2(\omega), f\_2(\omega^2), f\_2(\omega^3), f\_2(\omega^4)} \\
\mathsf{Enc}(a\cdot f\_1 + b \cdot f\_2) =
\Big{
\begin{aligned}
a\cdot f\_1(\omega) + b\cdot f\_2(\omega), a\cdot f\_1(\omega^2) + b\cdot f\_2(\omega^2), \ a\cdot f\_1(\omega^3) + b\cdot f\_2(\omega^3), a\cdot f\_1(\omega^4) + b\cdot f\_2(\omega^4)
\end{aligned}
\Big}
$$

The traditional encoding function used in RS-codes involves FFT operations. However, due to the limitations mentioned earlier, alternative encoding methods must be explored.

> Written by [Ryan Kim](mailto:undefined) of Fractalyze


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