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The following sequences work as $\Sigma$ for $f(x)=x^2$ $x_n=2^n-1$ has Hamming weight $n$ as does $x_n^2=2^{2n}-2^{n+1}+1$ (the number $2^{2n}-2^{n+1}$ has a run of $n-1$ ones followed by $n+1$ zer …

A thing to remember is that the customer is interested in the probability of correct reception after the error-correcting-code has done its magic. The 5-dimensional Reed-Muller code of length 16 and …

This is basically a retranslation of the problem into algebraic language. We are given an $n\times(n+k)$ matrix $A$ with entries in $GF(2)$ of the form $A=(I_n\mid B)$, where the matrix $B$ has no zer …

As an example I describe what is known about the BCH-codes for the purposes of this question. Just to give an idea of the type of results that are known. I suspect that they are not very useful for th …

You do know how to calculate the minimum distance (=free distance?) of a convolutional code? Cut the first edge from the zero state to the zero state (to disallow the all zeros word), and run Viterbi …

As pointed out by Gerry Myerson, this has a lot to do with autocorrelation of a sequence. More specifically incomplete (or aperiodic) autocorrelation. What I say below will give something non-trivial …

The commenters got it right. The automorphism group of a binary code is the set of permutations of coordinates that stabilizes the code. If instead of using the binary alphabet we use a ternary, quate …

A bit of terminology: An error-detection-code usually means something else. An error-detection-code is expected to raise a flag if something is wrong, IOW if the received sequence is not a valid code …

Your list certainly has many nice topics. 1) Yup. This would be nice to have. In practical applications we can get rid of the error-floor by concatenating a decent LDPC with a good high rate algebr …

The answer depends on several things. If your channel (or receiver) produces erasures and errors, then the relevant metric is the Hamming metric, as a code with minimum distance $d$ can correct a com …

This is, indeed, an open question for most values of $n$. A.D. Brouwer maintains a database of the best known lower and upper bounds, and everything that is known for small $n$ can be found there. A m …

The following simple (suboptimal) bound will get you started. It is easier to give a lower bound for the degree $n$ of the polynomials in terms of the minimum distance $d$, so I will do that. Feel fre …

Won't the following argument show that the difference between successive exponents can never be bounded away from zero no matter how clever you try to be in selecting $(a,b)$? The idea is to consider …

Presumably you forgot to add the condition $\gcd(g_1(x),g_2(x))=1$ for otherwise you would allow catastrophic encoders (a finite number of channel errors may cause an infinite number of errorneously i …

Adding a few words from the coding theory side. Abelian groups without $p$-torsion are somewhat more natural in coding theory, because then we get the machinery of discrete Fourier transform (which is …