# Tagged Questions

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### “Codes” in which a group of words are pairwise different at a certain position

I read the following problem, claimed to be in the IMO shortlist in 1988: A test consists of four multiple choice problems, each with three options, and the students should give an unique answer ...
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### Cyclic Hamming Code

I know that Hamming codes can be arranged in cyclic form. But my question is how can I prove this. My idea was to find a generator/primitive polynomial $p(x)$? For example I want to show that the ...
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### Reference for partial Hadamard matrices

Definition. An $m\times n$ matrix is said to be a partial Hadamard matrix (let's say PHM) if its entries are chosen from $\lbrace -1, 1 \rbrace$ such that the dot product of each pair of row vectors ...
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### When does the Lloyd polynomial have only integral roots?

For a $t$-error correcting code of length $n$ over the finite field $\mathbb{F}_q$, the Lloyd polynomial is given by $$L_t(n,x):=\sum_{j=0}^t(-1)^j\binom{x-1}{j}\binom{n-x}{t-j}(q-1)^{t-j}.$$ A ...
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### Extended Hypercube Graph

Definition 1. The $n$-hypercube graph has vertices which are the elements of the set $\lbrace 0,1\rbrace^n$ of $n$-bit binary strings, and an edge is drawn between each pair of vertices representing a ...
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### Bounded Hamming distance

Definition 1. For each $n\in\mathbb{Z}^+$, the $n$-dimensional Hamming cube is the set of ordered $n$-tuples of $\lbrace 0,1\rbrace$, denoted by $\lbrace 0,1\rbrace ^n$. Definition 2. The binary ...
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### Hamming codes from overlapping vectors

I am interested in whether the following problem is known. For a given binary vector $V$ of length $n\geq m$, let $S$ be a subset of the possible subvectors of $V$ of length $m$ and say that the size ...
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### Best upper bound on rate for q-ary codes

Among the many upper bounds for families of codes in $\mathbb F _2 ^n$, the best known bound is the one by McEliece, Rodemich, Rumsey and Welch which states that the rate $R(\delta)$ corresponding to ...
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### Automorphisms of subgroup of hamming cube under distance constraint

Let $S$ be a subset of $\{0,1\}^n$ such that any two elements of $S$ are at least (Hamming) distance 5 apart. I'm looking for an upper bound on the size of the automorphism group of $S$. There's a ...
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### A search for optimal order ideals

At the behest of Gerhard Paseman I'll describe the problem that I alluded to in name for a partial order. Let $M = M(\infty)$ denote the set of all finite subsets of the positive integers ...
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### Probability of false decoding with LDPC codes

It is well known that given a binary linear code $C$ under maximum likelihood decoding the probability of false decoding $P_C$ in a Binary Symmetric Channel with cross over probability $p$, in other ...