A code $C$ is said to be an $F_t$-linear code over $F_q$ if $C$ is a subspace of the $F_t$-vector space $F^{n}_{q}$ where $q = t^2$ is a prime power and $n$ is the length of the codeword.
Let us say that $t = 2$ i.e., consisting of the alphabets $\{0, 1\}$ and $q = 2^2 = 4$ i.e., consisting of the alphabets $\{0, 1, \omega, \omega^{2}\}$.
My questions are (the most important question is the first one):
- What is meant by $F_t$-linear code over $F_q$?
- What are the properties of $C$ (minimum weight, dimension, etc)?
- Will $C$ always consist of the alphabets $\{0, 1\}$? If so, does that mean that all linear binary codes are $F_2$-linear code over $F_4$?
- How about for general $t$ and $q$ satisfying $q = t^2$ and $t$ is a prime power?
A linear code over a finite field $\ F_q\ $ is just a linear subspace $\ C\ $ of $\ F_q^n\ $ for some positive integer $\ n\ $. This means that the vector sum $\ c_1+c_2\ $ of any two codewords $\ c_1,c_2\in C\ $ is also a codeword, and the scalar multiple $\ fc\ $ of any codeword $\ c \in C\ $ by any scalar $\ f\in F_q\ $ is also a codeword.
If $\ F_t\ $ is any subfield of $\ F_q\ $ (which means that $\ q=t^r\ $ must hold for some positive integer $\ r\ $), then any linear subspace of $\ F_q^n\ $ of dimension $\ k\ $ will be a vector space of dimension $\ kr\ $ over $\ F_t\ $. A subset $\ C\ $ of $\ F_q^n\ $ is an $\ F_t$-linear code over $\ F_q\ $ if it is a vector space over $\ F_t\ $. This means that the vector sum $\ c_1+c_2\ $ of any two codewords $\ c_1,c_2\in C\ $ is also a codeword, and the scalar multiple $\ fc\ $ of any codeword $\ c \in C\ $ by any scalar $\ f\in F_t\ $ is also a codeword. There is no requirement here that $ r=2\ $(i.e. that $\ q=t^2\ $), and it is possible that $\ C\ $ is not a $\ F_q$-linear subspace of $\ F_q^n\ $, because it is not necessarily true that $\ C\ $ is closed under scalar multiplication by an arbitrary member of $\ F_q\ $, only under multiplication by those that belong to $\ F_t\ $.
By definition, the alphabet of a code over a set $\ \Sigma\ $ is just the set $\ \Sigma\ $ itself. Thus, the alphabet of an $\ F_t$-linear code over $\ F_q\ $ is $\ F_q\ $, not $\ F_t\ $. In particular, the alphabet of an $\ F_2$-linear code over $\ F_4=\left\{0,1,\omega,\omega^2\right\}\ $ is $\ \left\{0,1,\omega,\omega^2\right\}\ $, not $\ \{0,1\}\ $. It is true that any linear binary code will be isomorphic as an $\ F_2$-vector space to some $\ F_2$-linear code over $\ F_4\ $, but since the alphabets of the two codes are different, they can't be considered identical as codes.