# Coding Theory: A code $C$ is an $F_t$-linear code over $F_q$?

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?
• What does your definition of $F_t$-linear mean when $t=3$ and $q=4$? Absent any stated relationship between $q$ and $t$, the notion of $F_t$-linear doesn't make much sense. Apr 10, 2020 at 22:48
• @DilipSarwate, I forgot to add the condition $q = t^2$. Sorry! Apr 10, 2020 at 23:04

## 1 Answer

• 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.

• Thank you for the response, would it be possible if you can kindly suggest any scholarly paper/book that could help me out further on this? Apr 23, 2020 at 9:11
• " ... and it is possible that $\ C\$ is not a $\ F_q$-linear subspace of $\ F_q^n\$ ..." should it be "... and it is possible that $\ C\$ is *not* a $\ F_t$-linear subspace of $\ F_q^n\$..."? Apr 23, 2020 at 10:01
• No. If $\ F_t\$ is a proper subfield of $\ F_q\$ then any vector space over $\ F_q\$ will be a vector space over $\ F_t\$, but a vector space over $\ F_t\$ will not necessarily be a vector space over $\ F_q\$, because it will not necessarily be closed under scalar multiplication by arbitrary elements of $\ F_q\$. Apr 23, 2020 at 13:33
• Taking your example with $\ t=2\$ and $\ q=4\$, for instance, the set $\ C=\{(a, b\omega)\,|\,a,b\in F_2\}\subseteq F_4^2\$ is $\ F_2$-linear but not $\ F_4$-linear because it is not closed under scalar multiplication by $\ \omega\$ or $\ \omega^2\$ ($\ (1,0)\in C\$ but $\ \omega(1,0)\not\in C\$ and $\ \omega^2(1,0)\not\in C\$). A specialist in coding theory (which I am certainly not) would be better able to suggest a suitable text. I was introduced to the subject by Elwyn Berlekamp's now classic text Algebraic Coding Theory, which is an excellent book which you might find useful. Apr 23, 2020 at 13:39
• Thank you so much for all of this! Apr 24, 2020 at 3:31