Embedding The Quotient Code

Question

A quotient code is the linear quotient $\frac {B}{A}$ of two binary linear codes equipped with the usual quotient norm $||\space||$ defined by $||b+A||=min_{a\in A}{||b+a||}$. Show that for some $n\in \Bbb N$ the quotient code $\frac {B}{A}$ is embedded in $\Bbb F_2^n$ via linear isometry.

Edit (lifted from a comment, JL): The term "quotient code" given here is being defined here – not imported from some other unnamed source to the very best of my knowledge. Quotient spaces are useful in coding theoretic arguments, and have been used by many including Slepian. The idea was a simple late night musing in coding theory

Answer 1

The claim is patently false. For a counterexample let $B$ be the universal code of length $7$, and let $A$ be the $(7,4,3)$ Hamming code.

Then:

• Because $A$ is a perfect single-error-correcting code, every non-trivial coset $b+A$, $b\notin A$ has a weight one coset leader.
• By the previous bullet, an isometry $\phi:B/A\to\Bbb{F}_2^m$ for some $m$ must map all the non-trivial cosets $b+A$ to words of Hamming weight one.
• If $b_1+A$ and $b_2+A$ are distinct non-trivial cosets of $A$, then $b_1+b_2+A$ is another, and should also be mapped to a word of Hamming weight one.
• But it is impossible for all three vectors $\phi(b_1+A)$, $\phi(b_2+A)$ and $\phi(b_1+b_2+A)$, the last of which must be equal to $\phi(b_1+A)+\phi(b_2+A)$, to be of weight one. That sum of two words of weight one has an even weight.