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Hadamard code is indeed the answer you want. Hadamard matrices are conjectured to exist for all n which are multiples of 4. This is very hard to prove and a longstanding open problem. Sylvester Hadama …

Your question, as explained in your last comment, has no answer in general. Due to transformations which yield equivalent codes, it is unlikely to have a short description of columns for every optima …

Getting a matrix with more than $q$ columns seems to be problematic. However taking, for example, a $q\times q$ Cauchy matrix would work for $n\leq q$. A Cauchy matrix is an m×n matrix with elements …

This is a problem that coding theorists have been considering since the late 1950's, both for BCH, and more generally for cyclic codes. The problem is combinatorial in nature, tightly bound to polyno …

Well, if the function $f$ has range $GF(2)^m$, represented by $GF(2^m)$ if convenient, it has rate 1/2. Such a function can really control symbol ($GF(2^m)$ ) not bit errors so it is a code over $GF(2 …

The codeword $c(\beta)$ in your question is defined on a set via the trace map that does not include the zero element. Thus its number of nonzero elements is simply the length minus the number of $k\i …

In Weight Divisibility of Cyclic Codes, Highly nonlinear functions on $F_{2^m}$, and crosscorrelation of maximum length sequences, Canteaut, Charpin, and Dobbertin (Siam J. Discrete Math. 13(1):105-13 …

In this paper some useful information is given, see the introduction and references. The problem of finding the weights is NP-complete but not known to be NP-hard. Without the generating matrix, the b …

The term direct sum of codes $C_1$ and $C_2$ is used for the code $$\{(u|v): u \in C_1, v \in C_2\},$$ and using this term in a search results in some publications. Ruud Pellikaan uses the term Juxta …

Interesting question. I hope I haven't misunderstood it but the discussion on p. 315 of Handbook of Coding Theory, Vol. 1, by Brouwer seems to imply that a finite field is necessary since it relies o …

Variable Length Error Correction (VLEC) codes do exist, and they are used in some applications where source coding/channel coding separation may not be the best way to go and both compression and erro …

Your notation is nonstandard. Code length is $n,$ dimension is $k,$ minimum distance is $d=n-k+1$ when a code is MDS. The paper by Alderson available here proves the following: Theorem 2. A $q-$ary $ …

Consider the Reed Solomon code with parameters $[n,k,d=n-k+1]$ over the field $\mathbb{F}_q.$ Take $q=2^m,$ and note that the usual choice is to choose $n=q-1,$ for $q-$ary symbols. Therefore the bina …

There is a nice overview of this problem in the beginning of the following paper by Sendrier and Simos which is a good place to start. Essentially the problem is harder than Graph Isomporphism (see p …

For a binary code, as $n$ grows you cannot do better than the repetition code $$C=\{11\cdots1,00\cdots0\},$$ with two codewords as soon as the minimum distance required $d>n/2.$ See Theorem 4 of Venka …