# Tagged Questions

error-correcting codes, error-detecting codes and related algebraic and/or combinatorial constructions

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### What is a divisor (of an algebraic curve)?

So if I have a polynomial $p(x,y)$ and define a curve $C$ based on $p$, what is a divisor? In the context I'm looking at (where I'm trying to learn about Goppa codes), in Joyner et al.'s "Applied ...
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### $C$ is a Reed-Solomon code. $1_n \in C$ iff $g(1)\neq 0$.

I have no idea how to prove the following. Could you help me? Let $C$ be a Reed-Solomon code of length $n$ and let $g$ be its generating polynomial. $1_n \in C$ iff $g(1)\neq 0$, where $1_n$ is a ...
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### BCH Coding: Parity Check Matrix

I was referring to a publication on BCH codes in GF(64) (http://ipnpr.jpl.nasa.gov/progress_report2/42-38/38O.PDF) where I noticed that the Parity Check Matrix (attached image) that they have derived ...
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### Probability of a word being incorrectly decoded.

Let C be the ternary repetition code of length 4 over the alphabet $\{0,1,2\}$. If the probability of each symbol being wrongly received is t and each symbol is likely the probability of a word being ...
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### How to solve binary equation which has mod?

Three messages in binary format are sent $$a_0 a_1 a_2 a_3$$ and coded in binary format $$b_0 b_1 b_2 b_3 b_4 b_5 b_6$$ Symbols $$b_0,b_1,b_2,b_3,b_4,b_5,b_6$$ are the coefficients of the Boolean ...
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### Understanding the hamming bound

I have a theorem for the hamming bound or the sphere packing bound. A q-ary $(n, m, 2e+1)$ code satisfies $$M \bigg\{ \binom {n}{0} + \binom{n}{1} (q-1)+...+\binom{n}{e}(q-1)^e\bigg\} \leq q^n$$ ...
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### The probability of i errors in specified positions

In a binary code of length n p(exactly i errors in specified positions)=$t^i (1-t)^{n-i}$. I have an example where $C=\{000,111\}$, the binary repetition code of length 3. Suppose 111 is ...
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### Dimension of BCH code of length 80 [closed]

I want to answer this question from Rudolf applied algebra. Determine the dimension of a 5-error-correcting BCH code over $F_3$ of length 80. Is there a fast way to do this ?
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### degree of generator polynomial $m(x)$

Suppose $Q$ is cyclic $(h,q)$ code over $F_u$ such that $\gcd(h,u) = 1$. Prove that degree of the generator polynomial $m(x)$ of $Q$ is $h - q$. Why do we need the condition $\gcd(h,u) = 1$ ? Any ...
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### Does generator matrix of a code $C$ must have linearly independent rows?

Def: A generator matrix $G$ with entries in $\mathbb{F}_q$ generates code $C$, and each rows of $G$ is basis of $C$. Does generator matrix of a code $C$ must have linearly independent rows? ...
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### Minimize the rank of a matrix with some entries known

Let $m,n$ be two positive integers, with $m\geq n$. Suppose we have $m$ sets $A_1,\ldots, A_m\subseteq [n]$, with $|A_i|=d_i$. Let $\mathbb F$ be a finite field of size $q$. Let $D$ be the set ...
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### Why extend the perfect binary Golay code?

The perfect Golay code [23,12,7] is most often seen in its extended version [24,12,8], with the added parity bit. The extended Golay code has had a lot of practical applications. But why not the ...
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### Word lengths of optimal binary code

Given an optimal binary code (ie the expected word length if as small as possible while the code is still decipherable) with word lengths $s_1, \ldots,s_m$, I'd like to show the following ...
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### Power moments identities for homogeneous weight

Pless power moments identities for the Hamming distance are a far reaching consequence of the MacWilliams identities. For codes in the homogeneous metric is there an analogue of the former when the ...
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### Coding Theory Proof, prove C is m-1 Error - Correcting but not m error correcting

Let A be a q-ary alphabet where q ≥ 2. Let m ∈ N and let C be the repetition code of length 2m over A Prove that C is (m−1)-error correcting, but not m-error correcting So far I know that something ...
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### trouble understanding calculation of signal-to-noise for ldpc codes

My apologies if the answer to this question is too easy. I am a mathematics student and the subject of low density parity check codes is new to me. In many papers on LDPC codes, there are plots ...
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### number of strings with hamming distance exactly $d$

Given a set $S=\{x_1,x_2,\dots,x_n\}$ each $x_i$ can take a value from $\{0,1,2,\dots,k\}$ and a distance $d>0$, Let $D$ be the set of vectors that are pairwise different from each other by exactly ...
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### Need help with the following problem on the decoding of BCH codes?

I have to write down a quasi parity check matrix of a BCH code of lenth 63 over $\mathbb{F}_{2^2}=\{0, 1, a, b=1+a\}$, then decode the vector ...
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### Distance $d$-independent set in hypercube

Given a graph $G = (V, E)$, a distance $d$-independent set is a subset $S \subseteq V$ such that any two vertices $x, y \in S$ have distance at least $d$. Thus traditional independent sets are ...
Introduction: Suppose $C$ is an $\left [ n,k \right ]$ code. Let $I_{k}$ be the $k\times k$ identity matrix. Let $P$ be a $k\times \left (n-k \right )$ matrix. Then, $\left ( I_{k} | P\right )$ is ...