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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?
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  • $\begingroup$ 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. $\endgroup$ Apr 10, 2020 at 22:48
  • $\begingroup$ @DilipSarwate, I forgot to add the condition $q = t^2$. Sorry! $\endgroup$ Apr 10, 2020 at 23:04

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  • 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.

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  • $\begingroup$ 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? $\endgroup$ Apr 23, 2020 at 9:11
  • $\begingroup$ " ... 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\ $..."? $\endgroup$ Apr 23, 2020 at 10:01
  • $\begingroup$ 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\ $. $\endgroup$ Apr 23, 2020 at 13:33
  • $\begingroup$ 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. $\endgroup$ Apr 23, 2020 at 13:39
  • $\begingroup$ Thank you so much for all of this! $\endgroup$ Apr 24, 2020 at 3:31

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