Questions tagged [coding-theory]

Use this tag for questions about source-coding and channel-coding in information theory, error-correcting codes, error-detecting codes, and related algebraic and/or combinatoric constructions. This tag should not be used for questions about programming.

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Prove that binary non-linear code with parameters (9,51,3) doesn't exist?

I have a problem with solving the following question : Prove that binary non-linear code with parameters (9,51,3) doesn't exist. Can somebody reveal what would be the approach there?
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Determine if a generator matrix G' also generates the same code C (generated by generator matrix G)

I am asked that if I know that a binary code $C$ is generated by a matrix $G$, how to show that another matrix $G'$ does or does not generate that same code $C$. I have deduced the parity matrix $P$ ...
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Any binary linear code of block length $4$ do not attain the hamming bound

There does not exist any binary linear code with block length $4$ that achieves the hamming bound. I am unable to proceed at all. Please provide me some hint.
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Number of $1$'s in a generator matrix?

Let $G$ be a generator matrix of an $[n,k,d]$ code. Then $G$ has atleast $kd$ many $1's$ in it. Please give me some hint.
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how to construct a binary matrix with given row and column distribution

I need to construct an $m \times n$ binary matrix $B$ from a given row and columns distribution. What algorithms can be used for this? As a concrete example : $B$ a $12 \times 63$ matrix with row ...
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The Entropy of the quantization symbols and the smallest number of bits that is required for representing source symbols.

Considering $N_s$ source symbols $v$ with PDF given as, $p(v)= e^{-v}I_{[0,\infty]}(v)$ The $k$-bits quantizer $Q$ maps the $N_s$ source symbols $v$ to symbols $s$, $s \in \{0,..., 2^k-1\}$ The ...
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Splitting fields- Coding Theory

I recentelly started to study about coding theory and I am having a hard time to understand the meaning of spliting fields over finite fields. For example: Q1 :what is the spllitting field of $X^9+1$ ...
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What is The highest number of information bits $N_b$ and the smallest number of bits?

I want to find the highest number of information bits $N_b$ that can be reliably transmitted over a binary symmetric channel (BSC) with fixed channel parameters, error probability $\sigma$ = range ...
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Hamming Distance Metric Error correcting code tree structure

Recently was reading about Error Correcting Codes & Hamming distance Metric in Introduction to Topology, Adams and Fransoza book and how a topological Open Ball of radius $r$ is placed around a ...
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Light bulbs in a high dimensional grid

Consider a $n \times n \times n$ array of light bulbs. In each step, one can flip lights from on to off and off to on along a 1d row in the $x$-, $y$- or $z$-axis. Suppose one has a configuration that ...
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BCH code $F^{q}_{2}$ field

I'm trying to understand the algorithm of construction the bch code. Polynomial $m_{\alpha}(x) \in F_{2}[x]$ is called the minimal polynomial for an element $\alpha \in F^{q}_{2}$ if it is an ...
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Hamming distance equals hamming weight under $XOR$ closure

I encountered the following question: Given a set $S$ of binary strings, each one contains $n$ bits, we define the weight of the set, $w(S)$ as the minimal hamming weight of a non zero string in $S$ (...
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From Automorphism of code to automorphism of lattice

From a code, a lattice can be constructed using many methods. For codes over $\mathbb{F}_2$, there is the straight construction \begin{equation} \Lambda(C) := \{v/\sqrt{2} \ | \ v \in \mathbb{Z}^n ...
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Can coefficients of a weight enumerator ever be non-unimodal?

Let C be a linear code and W(C) its weight enumerator with $W(C)=1+a_dx^d+...+a_{n-d}x^{n-d}+a_nx^n.$ Computations always show that the a_i's are ascending up to the middle (then descending). My ...
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Single-Parity-Check Codes

What are single parity-check(SPC) codes? I know about Repetition codes and generator matrix for them but have not been able to find much information about the SPC on the internet. Can anyone suggest ...
What is the standard representation for items in the ring $\mathbb F_q[x]$ where $q$ is a power of a prime?
I know that when $K$ is a field (or even more generally a ring) then $K[x]$, the set of all the polynomials of one variable $x$ whose coefficients are in $K$, is a ring itself (with sum and product &...