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# Questions tagged [coding-theory]

The mathematical theory of codes, as used in communication, data compression, and cryptography.

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16 views

### Shannon's Code Ordering of Probabilities

The first step in the Shannon's Code is to order the probabilities in decreasing order. Why do we do this? Is it to make it prefix-free or assign most probable words with shorter codewords?
89 views

### Testing if a distribution over $\mathbb{F}_2^n$ is heavily supported on a subspace

Let $P$ be a distribution over n-bitstrings which we will view as elements of $\mathbb{F}_2^n$. Given sample access to $P$, I am looking for an algorithm that tests if $P$ is heavily concentrated on a ...
1 vote
78 views

### Detecting Erroneous Corrections

A block code $C$, with minimum distance $d$ can be used to: Detect $d - 1$ errors Correct $\lfloor\frac{d - 1}{2}\rfloor$ errors However, the above usually assumes that the number of errors that are ...
87 views

### Application of PCP and error correcting codes to LLMs?

Are there any interesting results in applying error correcting codes and ideas from PCP (Probabilistically Checkable Proofs) to improve the quality of large language models (LLM), or connecting them ...
1 vote
100 views

### Code indistinguishability assumption for Code based cryptography (in special cases)

Cryptosystems that are based on error correcting codes are often based with hardness of the two problem. Computational syndrome decoding is hard Indistinguishability Assumption (IA): Distinguishing ...
98 views

### The existence of (non-rectangular) two dimensional Gray code

A Gray code consists of $n$-bit distinct strings $s_1,s_2, \ldots,s_N$ such that each $s_i$ and $s_{i+1}$ differs by one bit. For example: $000, 001, 010, 011, 111, 110, 101, 100$. It is known that we ...
87 views

### Existence of a family of size 2^Ω(n) of subsets of {1,...,n} each of cardinality n/4 where two subsets have at most n/8 elements in common

Let $\mathcal{G}$ be a family of $t=2^{\Omega(n)}$ subsets of $N=\{1,2,...,n\}$, each of cardinality $n / 4$ so that any two distinct members of $\mathcal{G}$ have at most $n / 8$ elements in common. ...
214 views

### Are all linear-rate and -distance classical linear codes expanding?

Consider a LDPC linear code defined as $\ker H$ for a $O(1)$ row- and column-sparse matrix $H \in \{0,1\}^{n \times r}$ with independent rows. Assume the code is linear-rate meaning $n - r = \Omega(n)$...
185 views

My question is about the following result about list-decoding of Sudan, Trevisan and Vadhan. (The formulation is taken from Shuichi Hirahara's paper.) I do not understand how this is possible. I ...
1 vote
20 views

### Coloring the $k$-deletion graph “constructively”

For $n,k\ge 1$, we define the graph $D_{n,k}$ to have vertex set $\{0,1\}^n$, with distinct $x,y$ being adjacent if $LCS(x,y)\ge n-k$. My question is: fixing $k>1$, does there exist some $C=C_k$ ...
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### Quantum security of cryptosystems: Are any non-Goppa code-based systems resistant to hidden subgroup attacks?

One of the main candidates for post-quantum cryptography is code-based cryptography (as opposed to lattice-based). The Niederreiter cryptosystem based on Goppa codes is shown to be resistant to hidden ...
88 views

### Approximate (in hamming distance) subset representation

Let us have a set $S$ and a subset $T \subseteq S$. I want to find an approximate representation of $T$, i.e. I want to represent (exactly) a set $T'$ that is close to $T$. That is, I want the ...
160 views

### Is the following graph an expander graph?

Let's say we have the following bipartite-graph, denoted $G=(L,R,E)$: It has the following adjacency matrix: I am having problems decoding a received word from what I was told is an expander code ...
236 views

### Reference request for linear algebra over GF(2)

I have been looking for materials on the linear algebra over $GF(2)$ but so far I haven't found any substantial textbooks or notes on this subject. In fact in one of the notes I found the introduction ...
1 vote
75 views

### Non-random errors with a Reed Solomon code

If I have a RS code, say [46, 16, 31], then I have a guaranteed error correction up to 15 symbols. I have no idea if it matters, but the code I have in front of me ...
362 views

### Information and Coding Theory Texts

I am coming from a pure mathematics (in analysis) background and am curious to learn some information and coding theory. I am after some recommendations on texts. Due to my personal background I am ...
1 vote
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### Is there a standard interpretation of this quantity?

Define an encoding scheme by a pair of functions: $$\mathsf{encode} : \mathcal{M}\to\mathcal{X},\quad \mathsf{decode} : \mathcal{X}\to\mathcal{M}$$ Typical examples of encoding schemes are things like ...
165 views

### What are the general direction and target question in the field of quantum error correction?

After quantum error correction was introduced in mid '90s, in subsequent years many of the classical analogues regarding the structure of code (such as singleton bound, GV bound etc) were obtained in ...
240 views

### Survey on Quantum error correction

Are there any standard recent survey articles on quantum error correction (and may be including fault Tolerant computing)? The most standard ones that many people refer to are this and this. Both of ...
200 views

### Damerau–Levenshtein distance with transposition of non-adjacent characters?

Wondering if it's possible to calculate Damerau–Levenshtein distance with transposition of non-adjacent characters (DL distance allows transposition of immediately adjacent characters only). I want ...
207 views

### Explicit Bits-back Coding (a.k.a. Free Energy Coding) applied to Gaussian mixtures

I've been trying to understand Bits-back coding (Frey, B. J., and G. E. Hinton. 1997.) a bit more (pun intended), which can be used to encode data with latent variable models. This tutorial by Pieter ...
1 vote
133 views

### Quantum error correction and graph codes

I was reading combinatorial approach towards quantum correction. A lot of work in this is on finding diagonal distance of a graph. Let me add definition of diagonal distance so that this remains self-...
413 views

### Minimum distance of a code

Is there a way to compute minimum distance of a code given a systematic parity check matrix? I know that min dist is smallest number $d$ such that there exists $d$ linearly dependant columns. I am ...
209 views

### Why Asymptotic Equipartition Property theorem proofs assume the source is memoryless?

I do not understand the assumption $X_1, X_2, \cdots$ are i.i.d. ~p(x) in the AEP proofs I have seen. I have read some different sources for understanding the Asymptotic Equipartition Property. Using ...
119 views

### Notation of sequences in rate distortion theory

I have been reading whatever sources I could get my hands on today, regarding this problem. Most notes online about rate distortion theory come from the book Elements of Information Theory by Thomas ...
112 views

### Relation between automorphism group of a linear code and its dual code

Are there any strong connections between automorphism groups of codes that are dual codes of each other? I am looking for statements like one charcterizes other or one gives bounds on other etc. In ...
1 vote
86 views

### Explicit binary codes with block length n, distance n / log n, rate 1 - o(1)?

What are the best (i.e., highest-rate) explicit binary codes with block length $n$ and minimum distance $d$, in the regime $d = n^{1 - o(1)}$? The "redundancy" of a code is the difference between the ...
55 views

### weights in low density codes

Generally, low density parity codes are decoded using sum product decoder (also known as decoding under belief propagation). Such codes are usually decoded nicely if there are no short length cycles ...
237 views

### Complexity of finding automorphism group of code

What is the computational complexity (may be both classical or quantum) for finding automorphism group of a general linear code? Is there better bound on complexity if structure of code is known for ...
478 views

### Bivariate low-degree polynomial testing of Polishchuk-Spielman

In the seminal paper of Polishchuk and Spielman where they give a construction of nearly linear sized $PCP$ for an $NP$ problem, one of the key ingredients is a low-degree test for bivariate ...
109 views

### How hard is it to approximate distance of linear code

I'm trying to figure out what is the current knowledge about how hard it is, given a generating matrix of a linear code over a field $F_{q}$, approximate it's distance. I of course found that ...
1 vote
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### How many different Huffman encoding for a given number of symbols

In Huffman coding, if we have two symbols to be encoded, we will get the result either 01 or 10. If we have three symbols, we ...
125 views

### Can Quarter-Subset Membership be decided space-efficiently?

Consider the following decision problem. Let $q = \sum_{i=0}^{n/4} \binom{n}{i}$ and let $(C_0^n, C_1^n,\dots,C_{q-1}^n)$ be a suitable enumeration of those subsets of $\{0,1,\dots,n-1\}$ that have at ...
893 views

### Explanation of polar decoding?

Polar decoding, according to Arikan's paper, can be decoded using a successive cancellation (SC) decoder, which is based on calculating the likelihood ratio, as seen in the image below. How can I ...
223 views

### Bounds for maximum number of code-words in a ternary error correcting code with length n and distance d?

I'm not sure I can find an explicit formula. Wondering if anyone can come up with lower/upper bounds.
### Explicit Formula of Delsarte's Linear Programming Upper Bound for $A_q(n,3)$
The problem of giving an explicit formula for $A_q(n,d)$ is sometimes referred to as "the main problem in coding theory." The value of $A_q(n,d)$ is given by the maximum number of codewords in a q-ary ...
In Coding Theory, people use $q$-ary alphabets: why do we need a finite set? Why can't we define codes over infinite sets. such as $\mathbb{R}$ or $\mathbb{C}$?