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You are given an i.i.d source with symbols taking value in the alphabet $\mathcal{A}=\{a,b,c,d\}$ and probabilities $\{1/8,1/8,1/4,1/2\}$. Consider making blocks of length $n$ and constructing a Huffman code that assigns a binary codeword to each block of $n$ symbols. Choose the correct statement regarding the average ... | In this example, the weighted average codeword length is 2.25 bits per symbol, only slightly larger than the calculated entropy of 2.205 bits per symbol. So not only is this code optimal in the sense that no other feasible code performs better, but it is very close to the theoretical limit established by Shannon. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
You are given an i.i.d source with symbols taking value in the alphabet $\mathcal{A}=\{a,b,c,d\}$ and probabilities $\{1/8,1/8,1/4,1/2\}$. Consider making blocks of length $n$ and constructing a Huffman code that assigns a binary codeword to each block of $n$ symbols. Choose the correct statement regarding the average ... | This leaves us with a single node and our algorithm is complete. The code lengths for the different characters this time are 1 bit for A and 3 bits for all other characters. This results in the lengths of 1 bit for A and per 3 bits for B, C, D and E, giving an average length of We see that the Huffman code has outperfo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
A bag contains the letters of LETSPLAY. Someone picks at random 4 letters from the bag without revealing the outcome to you. Subsequently you pick one letter at random among the remaining 4 letters. What is the entropy (in bits) of the random variable that models your choice? Check the correct answer. | In information theory, the entropy of a random variable is the average level of "information", "surprise", or "uncertainty" inherent to the variable's possible outcomes. Given a discrete random variable X {\displaystyle X} , which takes values in the alphabet X {\displaystyle {\mathcal {X}}} and is distributed accordin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
A bag contains the letters of LETSPLAY. Someone picks at random 4 letters from the bag without revealing the outcome to you. Subsequently you pick one letter at random among the remaining 4 letters. What is the entropy (in bits) of the random variable that models your choice? Check the correct answer. | Symmetrically, Player B presumes Player A expects the random numbers 2-100 and chooses a1, so B chooses b2. As a result, the players had a final result of (1, a2, b2), with a payoff of 1 for both - the lowest possible payoff (total or individual). Now suppose that it is common knowledge that the random number is 1 - th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Which of the following congruence classes has a multiplicative inverse? | In mathematics, particularly in the area of arithmetic, a modular multiplicative inverse of an integer a is an integer x such that the product ax is congruent to 1 with respect to the modulus m. In the standard notation of modular arithmetic this congruence is written as a x ≡ 1 ( mod m ) , {\displaystyle ax\equiv 1{\p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Which of the following congruence classes has a multiplicative inverse? | A reduced residue system modulo 10 could be {1, 3, 7, 9}. The product of any two congruence classes represented by these numbers is again one of these four congruence classes. This implies that these four congruence classes form a group, in this case the cyclic group of order four, having either 3 or 7 as a (multiplica... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Is $(\mathbb{Z} / 8\mathbb{Z}^*, \cdot)$ isomorphic to $(\mathbb{Z} / k\mathbb{Z}, +)$ for some $k$? | Finally, use that O {\displaystyle O} is a Dedekind domain and therefore each ideal can be written as a product of prime ideals. In other words, the map ( ⋅ ) {\displaystyle (\cdot )} is a K × {\displaystyle K^{\times }} -equivariant group homomorphism. As a consequence, the map above induces a surjective homomorphism ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Is $(\mathbb{Z} / 8\mathbb{Z}^*, \cdot)$ isomorphic to $(\mathbb{Z} / k\mathbb{Z}, +)$ for some $k$? | There is a natural ring homomorphism K 0 ( X ) → H 2 ∗ ( X , Q ) , {\displaystyle K^{0}(X)\to H^{2*}(X,\mathbb {Q} ),} the Chern character, such that K 0 ( X ) ⊗ Q → H 2 ∗ ( X , Q ) {\displaystyle K^{0}(X)\otimes \mathbb {Q} \to H^{2*}(X,\mathbb {Q} )} is an isomorphism. The equivalent of the Steenrod operations in K-t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider a $(k+1,k)$ block code that to a binary sequence $x_1,\dots,x_k$ associates the codeword $x_1,\dots,x_k,x_{k+1}$, where $x_{k+1}= x_1+\ldots+x_k$ mod $2$. This code can detect all the errors of odd weight. | We can improve this situation. If we use the generator polynomial g ( x ) = p ( x ) ( 1 + x ) {\displaystyle g(x)=p(x)(1+x)} , where p {\displaystyle p} is a primitive polynomial of degree r − 1 {\displaystyle r-1} , then the maximal total block length is 2 r − 1 − 1 {\displaystyle 2^{r-1}-1} , and the code is able to ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider a $(k+1,k)$ block code that to a binary sequence $x_1,\dots,x_k$ associates the codeword $x_1,\dots,x_k,x_{k+1}$, where $x_{k+1}= x_1+\ldots+x_k$ mod $2$. This code can detect all the errors of odd weight. | An exceptional block design is the Steiner system S(5,8,24) whose automorphism group is the sporadic simple Mathieu group M 24 {\displaystyle M_{24}} . The codewords of the extended binary Golay code have a length of 24 bits and have weights 0, 8, 12, 16, or 24. This code can correct up to three errors. So every 24-bit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
A generator matrix $G$ of binary $(6,3)$ linear code maps the information vectors $m_1 = (1,0,1)$ and $m_2=(1,1,1)$ into the codewords $c_1 = (1,1,0,0,0,1)$ and $c_2=(1,0,0,0,1,0)$ respectively. Which of the following is true? | If G is a matrix, it generates the codewords of a linear code C by w = s G {\displaystyle w=sG} where w is a codeword of the linear code C, and s is any input vector. Both w and s are assumed to be row vectors. A generator matrix for a linear q {\displaystyle _{q}} -code has format k × n {\displaystyle k\times n} , wh... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
A generator matrix $G$ of binary $(6,3)$ linear code maps the information vectors $m_1 = (1,0,1)$ and $m_2=(1,1,1)$ into the codewords $c_1 = (1,1,0,0,0,1)$ and $c_2=(1,0,0,0,1,0)$ respectively. Which of the following is true? | ExampleFrom the above matrix we have 2k = 24 = 16 codewords. Let a → {\displaystyle {\vec {a}}} be a row vector of binary data bits, a → = , a i ∈ { 0 , 1 } {\displaystyle {\vec {a}}=,\quad a_{i}\in \{0,1\}} . The codeword x → {\displaystyle {\vec {x}}} for any of the 16 possible data vectors a → {\displaystyle {\vec ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider a source $S$ with some distribution $P_S$ over the alphabet $\mathcal{A} = \{a, b, c, d, e, f\}$. Bob designs a uniquely decodable code $\Gamma$ over a code alphabet $\mathcal{D}$ of size $D$ with following codeword lengths. egin{center}egin{tabular}{ |c|c|c|c|c|c|c| } \hline& $a$ & $b$ & $c$ & $d$ & $e$ & $... | Let each source symbol from the alphabet S = { s 1 , s 2 , … , s n } {\displaystyle S=\{\,s_{1},s_{2},\ldots ,s_{n}\,\}} be encoded into a uniquely decodable code over an alphabet of size r {\displaystyle r} with codeword lengths ℓ 1 , ℓ 2 , … , ℓ n . {\displaystyle \ell _{1},\ell _{2},\ldots ,\ell _{n}.} Then ∑ i = 1 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider a source $S$ with some distribution $P_S$ over the alphabet $\mathcal{A} = \{a, b, c, d, e, f\}$. Bob designs a uniquely decodable code $\Gamma$ over a code alphabet $\mathcal{D}$ of size $D$ with following codeword lengths. egin{center}egin{tabular}{ |c|c|c|c|c|c|c| } \hline& $a$ & $b$ & $c$ & $d$ & $e$ & $... | Let C {\displaystyle {\mathcal {C}}} be a ( n , k , d ) q {\displaystyle (n,k,d)_{q}} error-correcting code; in other words, C {\displaystyle {\mathcal {C}}} is a code of length n {\displaystyle n} , dimension k {\displaystyle k} and minimum distance d {\displaystyle d} over an alphabet Σ {\displaystyle \Sigma } of siz... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider the following loaded dice with $6$ faces: $P(S_1=6)=5/6$ and $P(S_1 = x)=1/30$ for $x\in\{1,2,3,4,5\}$. Suppose we throw it indefinitely. Hence, we have a source $S=S_1 S_2 S_3\ldots$. Then, $H(S_n) = H(S_{n-1})$. | In general, if P(n) is the probability of throwing at least n sixes with 6n dice, then: P ( n ) = 1 − ∑ x = 0 n − 1 ( 6 n x ) ( 1 6 ) x ( 5 6 ) 6 n − x . {\displaystyle P(n)=1-\sum _{x=0}^{n-1}{\binom {6n}{x}}\left({\frac {1}{6}}\right)^{x}\left({\frac {5}{6}}\right)^{6n-x}\,.} As n grows, P(n) decreases monotonically ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider the following loaded dice with $6$ faces: $P(S_1=6)=5/6$ and $P(S_1 = x)=1/30$ for $x\in\{1,2,3,4,5\}$. Suppose we throw it indefinitely. Hence, we have a source $S=S_1 S_2 S_3\ldots$. Then, $H(S_n) = H(S_{n-1})$. | Here legal means that the coefficients are non-negative and sum to six, so that each die has six sides and every face has at least one spot. (That is, the generating function of each die must be a polynomial p(x) with positive coefficients, and with p(0) = 0 and p(1) = 6.) Only one such partition exists: x ( x + 1 ) ( ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider the following loaded dice with $6$ faces: $P(S_1=6)=5/6$ and $P(S_1 = x)=1/30$ for $x\in\{1,2,3,4,5\}$. Suppose we throw it indefinitely. Hence, we have a source $S=S_1 S_2 S_3\ldots$. Then, $H(S_1,\ldots,S_n) = \sum_{i=1}^n H(S_i|S_1\ldots S_{i-1})$. | In general, if P(n) is the probability of throwing at least n sixes with 6n dice, then: P ( n ) = 1 − ∑ x = 0 n − 1 ( 6 n x ) ( 1 6 ) x ( 5 6 ) 6 n − x . {\displaystyle P(n)=1-\sum _{x=0}^{n-1}{\binom {6n}{x}}\left({\frac {1}{6}}\right)^{x}\left({\frac {5}{6}}\right)^{6n-x}\,.} As n grows, P(n) decreases monotonically ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider the following loaded dice with $6$ faces: $P(S_1=6)=5/6$ and $P(S_1 = x)=1/30$ for $x\in\{1,2,3,4,5\}$. Suppose we throw it indefinitely. Hence, we have a source $S=S_1 S_2 S_3\ldots$. Then, $H(S_1,\ldots,S_n) = \sum_{i=1}^n H(S_i|S_1\ldots S_{i-1})$. | Here legal means that the coefficients are non-negative and sum to six, so that each die has six sides and every face has at least one spot. (That is, the generating function of each die must be a polynomial p(x) with positive coefficients, and with p(0) = 0 and p(1) = 6.) Only one such partition exists: x ( x + 1 ) ( ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider the following loaded dice with $6$ faces: $P(S_1=6)=5/6$ and $P(S_1 = x)=1/30$ for $x\in\{1,2,3,4,5\}$. Suppose we throw it indefinitely. Hence, we have a source $S=S_1 S_2 S_3\ldots$. Then, the source is stationary. | In general, if P(n) is the probability of throwing at least n sixes with 6n dice, then: P ( n ) = 1 − ∑ x = 0 n − 1 ( 6 n x ) ( 1 6 ) x ( 5 6 ) 6 n − x . {\displaystyle P(n)=1-\sum _{x=0}^{n-1}{\binom {6n}{x}}\left({\frac {1}{6}}\right)^{x}\left({\frac {5}{6}}\right)^{6n-x}\,.} As n grows, P(n) decreases monotonically ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider the following loaded dice with $6$ faces: $P(S_1=6)=5/6$ and $P(S_1 = x)=1/30$ for $x\in\{1,2,3,4,5\}$. Suppose we throw it indefinitely. Hence, we have a source $S=S_1 S_2 S_3\ldots$. Then, the source is stationary. | Here legal means that the coefficients are non-negative and sum to six, so that each die has six sides and every face has at least one spot. (That is, the generating function of each die must be a polynomial p(x) with positive coefficients, and with p(0) = 0 and p(1) = 6.) Only one such partition exists: x ( x + 1 ) ( ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider the following loaded dice with $6$ faces: $P(S_1=6)=5/6$ and $P(S_1 = x)=1/30$ for $x\in\{1,2,3,4,5\}$. Suppose we throw it indefinitely. Hence, we have a source $S=S_1 S_2 S_3\ldots$. Then, $\lim_{n o\infty}H(S_n) = \log_2(6)$. | In general, if P(n) is the probability of throwing at least n sixes with 6n dice, then: P ( n ) = 1 − ∑ x = 0 n − 1 ( 6 n x ) ( 1 6 ) x ( 5 6 ) 6 n − x . {\displaystyle P(n)=1-\sum _{x=0}^{n-1}{\binom {6n}{x}}\left({\frac {1}{6}}\right)^{x}\left({\frac {5}{6}}\right)^{6n-x}\,.} As n grows, P(n) decreases monotonically ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider the following loaded dice with $6$ faces: $P(S_1=6)=5/6$ and $P(S_1 = x)=1/30$ for $x\in\{1,2,3,4,5\}$. Suppose we throw it indefinitely. Hence, we have a source $S=S_1 S_2 S_3\ldots$. Then, $\lim_{n o\infty}H(S_n) = \log_2(6)$. | Here legal means that the coefficients are non-negative and sum to six, so that each die has six sides and every face has at least one spot. (That is, the generating function of each die must be a polynomial p(x) with positive coefficients, and with p(0) = 0 and p(1) = 6.) Only one such partition exists: x ( x + 1 ) ( ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider the group $(\mathbb{Z} / 23 \mathbb{Z}^*, \cdot)$. Find how many elements of the group are generators of the group. (Hint: $5$ is a generator of the group.) | the group elements, g, in G are functions of the parameters: and all parameters set to zero returns the identity element of the group: Group elements are often matrices which act on vectors, or transformations acting on functions. The generators of the group are the partial derivatives of the group elements with respec... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider the group $(\mathbb{Z} / 23 \mathbb{Z}^*, \cdot)$. Find how many elements of the group are generators of the group. (Hint: $5$ is a generator of the group.) | (Z/pZ)n is generated by n elements, and n is the least possible number of generators. In particular, the set {e1, ..., en} , where ei has a 1 in the ith component and 0 elsewhere, is a minimal generating set. Every elementary abelian group has a fairly simple finite presentation. ( Z / p Z ) n ≅ ⟨ e 1 , … , e n ∣ e i p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In RSA, we set $p = 7, q = 11, e = 13$. The public key is $(m, e) = (77, 13)$. The ciphertext we receive is $c = 14$. What is the message that was sent? (Hint: You may solve faster using Chinese remainder theorem.). | The public key in the RSA system is a tuple of integers ( N , e ) {\displaystyle (N,e)} , where N is the product of two primes p and q. The secret key is given by an integer d satisfying e d ≡ 1 ( mod ( p − 1 ) ( q − 1 ) ) {\displaystyle ed\equiv 1{\pmod {(p-1)(q-1)}}} ; equivalently, the secret key may be given by d p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
In RSA, we set $p = 7, q = 11, e = 13$. The public key is $(m, e) = (77, 13)$. The ciphertext we receive is $c = 14$. What is the message that was sent? (Hint: You may solve faster using Chinese remainder theorem.). | Setting up an RSA system involves choosing large prime numbers p and q, computing n = pq and k = φ(n), and finding two numbers e and d such that ed ≡ 1 (mod k). The numbers n and e (the "encryption key") are released to the public, and d (the "decryption key") is kept private. A message, represented by an integer m, wh... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider an RSA encryption where the public key is published as $(m, e) = (35, 11)$. Which one of the following numbers is a valid decoding exponent? | This RSA modulus is made public together with the encryption exponent e. N and e form the public key pair (e, N). By making this information public, anyone can encrypt messages to Bob. The decryption exponent d satisfies e d = 1 mod λ ( N ) {\displaystyle ed=1{\bmod {\lambda }}(N)} , where λ ( N ) {\displaystyle \lambd... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider an RSA encryption where the public key is published as $(m, e) = (35, 11)$. Which one of the following numbers is a valid decoding exponent? | If the public exponent is small and the plaintext m {\displaystyle m} is very short, then the RSA function may be easy to invert, which makes certain attacks possible. Padding schemes ensure that messages have full lengths, but additionally choosing the public exponent e = 2 16 + 1 {\displaystyle e=2^{16}+1} is recomme... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $K = (K_1, K_2,..., K_n)$, where each $K_i$ is $0$ or $1$ with probability $1/2$. Let $K'=(K'_1, K'_2, ..., K'_n)$ such that, for each $i$, $K'_i \in {0, 1}$ and $K'_{i} = \sum_{j = 1}^i K_j ext{mod} 8.$ True or false: Using $K'$ as the key one can achieve perfect secrecy if the message is $n$ bits. | Another way of stating perfect secrecy is that for all messages m 1 , m 2 {\displaystyle m_{1},m_{2}} in message space M, and for all ciphers c in cipher space C, we have Pr k ⇐ K = Pr k ⇐ K {\displaystyle {\underset {k\Leftarrow \mathrm {K} }{\operatorname {Pr} }}={\underset {k\Leftarrow \mathrm {K} }{\operatorname ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $K = (K_1, K_2,..., K_n)$, where each $K_i$ is $0$ or $1$ with probability $1/2$. Let $K'=(K'_1, K'_2, ..., K'_n)$ such that, for each $i$, $K'_i \in {0, 1}$ and $K'_{i} = \sum_{j = 1}^i K_j ext{mod} 8.$ True or false: Using $K'$ as the key one can achieve perfect secrecy if the message is $n$ bits. | The piling-up lemma allows the cryptanalyst to determine the probability that the equality: X 1 ⊕ X 2 ⊕ ⋯ ⊕ X n = 0 {\displaystyle X_{1}\oplus X_{2}\oplus \cdots \oplus X_{n}=0} holds, where the X's are binary variables (that is, bits: either 0 or 1). Let P(A) denote "the probability that A is true". If it equals one, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider a message $T$ and a key $K$ chosen independently from $T$. True or false: If there exists a perfectly secret encryption scheme using $K$, then $H(T) \leq H(K)$. | In a symmetric-key system, Bob knows Alice's encryption key. Once the message is encrypted, Alice can safely transmit it to Bob (assuming no one else knows the key). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider a message $T$ and a key $K$ chosen independently from $T$. True or false: If there exists a perfectly secret encryption scheme using $K$, then $H(T) \leq H(K)$. | A cryptosystem is considered secure in terms of indistinguishability if no adversary, given an encryption of a message randomly chosen from a two-element message space determined by the adversary, can identify the message choice with probability significantly better than that of random guessing (1⁄2). If any adversary ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $\mathcal{C}$ be a binary $(n,k)$ linear code with minimum distance $d_{\min} = 4$. Let $\mathcal{C}'$ be the code obtained by adding a parity-check bit $x_{n+1}=x_1 \oplus x_2 \oplus \cdots \oplus x_n$ at the end of each codeword of $\mathcal{C}$. Let $d_{\min}'$ be the minimum distance of $\mathcal{C}'$. Which of... | {\displaystyle \min _{c\in C,\ c\neq c_{0}}d(c,c_{0})=\min _{c\in C,\ c\neq c_{0}}d(c-c_{0},0)=\min _{c\in C,\ c\neq 0}d(c,0)=d.} In other words, in order to find out the minimum distance between the codewords of a linear code, one would only need to look at the non-zero codewords. The non-zero codeword with the smalle... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $\mathcal{C}$ be a binary $(n,k)$ linear code with minimum distance $d_{\min} = 4$. Let $\mathcal{C}'$ be the code obtained by adding a parity-check bit $x_{n+1}=x_1 \oplus x_2 \oplus \cdots \oplus x_n$ at the end of each codeword of $\mathcal{C}$. Let $d_{\min}'$ be the minimum distance of $\mathcal{C}'$. Which of... | Let B {\displaystyle {\mathcal {B}}} be a binary code consisting of M {\displaystyle M} codewords of length n {\displaystyle {\mathit {n}}} and minimum distance d min {\displaystyle {d_{\min }}} , such that c ∈ B {\displaystyle {\mathit {c}}\in {\mathcal {B}}} implies that c ¯ ∈ B {\displaystyle {\mathit {\bar {c}}}\in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $\mathcal{C}$ be a $(n,k)$ Reed-Solomon code on $\mathbb{F}_q$. Let $\mathcal{C}'$ be the $(2n,k)$ code such that each codeword of $\mathcal{C}'$ is a codeword of $\mathcal{C}$ repeated twice, i.e., if $(x_1,\dots,x_n) \in\mathcal{C}$, then $(x_1,\dots,x_n,x_1,\dots,x_n)\in\mathcal{C'}$. What is the minimum distanc... | In order to prove a lower bound for the distance of a code C ∗ {\displaystyle C^{*}} we prove that the Hamming distance of an arbitrary but distinct pair of codewords has a lower bound. So let Δ ( c 1 , c 2 ) {\displaystyle \Delta (c^{1},c^{2})} be the Hamming distance of two codewords c 1 {\displaystyle c^{1}} and c 2... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $\mathcal{C}$ be a $(n,k)$ Reed-Solomon code on $\mathbb{F}_q$. Let $\mathcal{C}'$ be the $(2n,k)$ code such that each codeword of $\mathcal{C}'$ is a codeword of $\mathcal{C}$ repeated twice, i.e., if $(x_1,\dots,x_n) \in\mathcal{C}$, then $(x_1,\dots,x_n,x_1,\dots,x_n)\in\mathcal{C'}$. What is the minimum distanc... | Furthermore, there are two polynomials that do agree in k − 1 {\displaystyle k-1} points but are not equal, and thus, the distance of the Reed–Solomon code is exactly d = n − k + 1 {\displaystyle d=n-k+1} . Then the relative distance is δ = d / n = 1 − k / n + 1 / n = 1 − R + 1 / n ∼ 1 − R {\displaystyle \delta =d/n=1-... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $\mathcal{C}$ be a binary $(6,3)$ linear code containing the codewords $\mathbf{x}_1 = 011011$, $\mathbf{x}_2 = 101101$ and $\mathbf{x}_3 = 111000$. True or false: The minimum distance of the code is $d_{\min} = 3$. | {\displaystyle \min _{c\in C,\ c\neq c_{0}}d(c,c_{0})=\min _{c\in C,\ c\neq c_{0}}d(c-c_{0},0)=\min _{c\in C,\ c\neq 0}d(c,0)=d.} In other words, in order to find out the minimum distance between the codewords of a linear code, one would only need to look at the non-zero codewords. The non-zero codeword with the smalle... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $\mathcal{C}$ be a binary $(6,3)$ linear code containing the codewords $\mathbf{x}_1 = 011011$, $\mathbf{x}_2 = 101101$ and $\mathbf{x}_3 = 111000$. True or false: The minimum distance of the code is $d_{\min} = 3$. | A code whose minimum distance is at least 3, have a check matrix all of whose columns are distinct and non zero. If a check matrix for a binary code has m {\displaystyle m} rows, then each column is an m {\displaystyle m} -bit binary number. There are 2 m − 1 {\displaystyle 2^{m}-1} possible columns. Therefore, if a ch... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $\mathcal{C}$ be a binary $(6,3)$ linear code containing the codewords $\mathbf{x}_1 = 011011$, $\mathbf{x}_2 = 101101$ and $\mathbf{x}_3 = 111000$. True or false: A generator matrix for the code is egin{equation*} G = egin{pmatrix} 1 &0 &0 &0 &1 &1 \ 0 &1 &0 &0 &0 &1 \ 0 &0 &1 &0 &1 &1 \end{... | From this, the generator matrix G can be obtained as {\displaystyle {\begin{bmatrix}I_{k}|P\end{bmatrix}}} (noting that in the special case of this being a binary code P = − P {\displaystyle P=-P} ), or specifically: G = ( 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 0 ) . {\displaystyle \mathbf {G} ={\begin{pmatrix}1&0&0&1&0&1\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $\mathcal{C}$ be a binary $(6,3)$ linear code containing the codewords $\mathbf{x}_1 = 011011$, $\mathbf{x}_2 = 101101$ and $\mathbf{x}_3 = 111000$. True or false: A generator matrix for the code is egin{equation*} G = egin{pmatrix} 1 &0 &0 &0 &1 &1 \ 0 &1 &0 &0 &0 &1 \ 0 &0 &1 &0 &1 &1 \end{... | If G is a matrix, it generates the codewords of a linear code C by w = s G {\displaystyle w=sG} where w is a codeword of the linear code C, and s is any input vector. Both w and s are assumed to be row vectors. A generator matrix for a linear q {\displaystyle _{q}} -code has format k × n {\displaystyle k\times n} , wh... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $\mathcal{C}$ be the $(6,3)$ linear code on $\mathbb{F}_3$ whose parity-check matrix is egin{equation*} H = egin{pmatrix} 2 &0 &1 &1 &1 &0 \ 1 &2 &0 &0 &1 &1 \ 0 &0 &0 &1 &1 &1 \end{pmatrix}. \end{equation*} True or false: The sequence $\mathbf{y} = 111000$ is a codeword of $\mathcal{C}$. | This is a (6, 3) linear code, with n = 6 and k = 3. Again ignoring lines going out of the picture, the parity-check matrix representing this graph fragment is H = ( 1 1 1 1 0 0 0 0 1 1 0 1 1 0 0 1 1 0 ) . {\displaystyle \mathbf {H} ={\begin{pmatrix}1&1&1&1&0&0\\0&0&1&1&0&1\\1&0&0&1&1&0\\\end{pmatrix}}.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $\mathcal{C}$ be the $(6,3)$ linear code on $\mathbb{F}_3$ whose parity-check matrix is egin{equation*} H = egin{pmatrix} 2 &0 &1 &1 &1 &0 \ 1 &2 &0 &0 &1 &1 \ 0 &0 &0 &1 &1 &1 \end{pmatrix}. \end{equation*} True or false: The sequence $\mathbf{y} = 111000$ is a codeword of $\mathcal{C}$. | In coding theory, a parity-check matrix of a linear block code C is a matrix which describes the linear relations that the components of a codeword must satisfy. It can be used to decide whether a particular vector is a codeword and is also used in decoding algorithms. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $\mathcal{C}$ be a binary $(5,2)$ linear code with generator matrix egin{equation*} G = egin{pmatrix} 1 &0 &1 &0 &1 \ 0 &1 &0 &1 &1 \end{pmatrix} \end{equation*} and consider a minimum-distance decoder obtained by choosing the coset leaders of the standard array of $\mathcal{C}$ so that th... | As a result, the half-the minimum distance acts as a combinatorial barrier beyond which unambiguous error-correction is impossible, if we only insist on unique decoding. However, received words such as y {\displaystyle y} considered above occur only in the worst-case and if one looks at the way Hamming balls are packed... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $\mathcal{C}$ be a binary $(5,2)$ linear code with generator matrix egin{equation*} G = egin{pmatrix} 1 &0 &1 &0 &1 \ 0 &1 &0 &1 &1 \end{pmatrix} \end{equation*} and consider a minimum-distance decoder obtained by choosing the coset leaders of the standard array of $\mathcal{C}$ so that th... | An example of a standard array for the 2-dimensional code C = {00000, 01101, 10110, 11011} in the 5-dimensional space V (with 32 vectors) is as follows: The decoding procedure is to find the received word in the table and then add to it the coset leader of the row it is in. Since in binary arithmetic adding is the same... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $b$ be the maximum number of linearly independent columns of a parity check matrix $H$ of a linear code. True or false: Then, the minimum distance of the code is $b+1$. | A code whose minimum distance is at least 3, have a check matrix all of whose columns are distinct and non zero. If a check matrix for a binary code has m {\displaystyle m} rows, then each column is an m {\displaystyle m} -bit binary number. There are 2 m − 1 {\displaystyle 2^{m}-1} possible columns. Therefore, if a ch... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $b$ be the maximum number of linearly independent columns of a parity check matrix $H$ of a linear code. True or false: Then, the minimum distance of the code is $b+1$. | Hence the rate of Hamming codes is R = k / n = 1 − r / (2r − 1), which is the highest possible for codes with minimum distance of three (i.e., the minimal number of bit changes needed to go from any code word to any other code word is three) and block length 2r − 1. The parity-check matrix of a Hamming code is construc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider the following mysterious binary encoding:egin{center} egin{tabular}{c|c} symbol & encoding \ \hline $a$ & $??0$\ $b$ & $??0$\ $c$ & $??0$\ $d$ & $??0$ \end{tabular} \end{center} where with '$?$' we mean that we do not know which bit is assigned as the first two symbols of the e... | For the example mentioned above, the encoding becomes: (1,1,2), ('B','A','C','D') This means that the first symbol B is of length 1, then the A of length 2, and remains of 3. Since the symbols are sorted by bit-length, we can efficiently reconstruct the codebook. A pseudo code describing the reconstruction is introduce... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider the following mysterious binary encoding:egin{center} egin{tabular}{c|c} symbol & encoding \ \hline $a$ & $??0$\ $b$ & $??0$\ $c$ & $??0$\ $d$ & $??0$ \end{tabular} \end{center} where with '$?$' we mean that we do not know which bit is assigned as the first two symbols of the e... | ASCII reserves the first 32 codes (numbers 0–31 decimal) for control characters known as the "C0 set": codes originally intended not to represent printable information, but rather to control devices (such as printers) that make use of ASCII, or to provide meta-information about data streams such as those stored on magn... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Suppose that you possess a $D$-ary encoding $\Gamma$ for the source $S$ that does not satisfy Kraft's Inequality. Specifically, in this problem, we assume that our encoding satisfies $\sum_{i=1}^n D^{-l_i} = k+1 $ with $k>0$. What can you infer on the average code-word length $L(S,\Gamma)$? | For Shannon's method, the word lengths satisfy l i = ⌈ − log 2 p i ⌉ ≤ − log 2 p i + 1. {\displaystyle l_{i}=\lceil -\log _{2}p_{i}\rceil \leq -\log _{2}p_{i}+1.} Hence the expected word length satisfies Here, H ( X ) = − ∑ i = 1 n p i log 2 p i {\displaystyle H(X)=-\textstyle \sum _{i=1}^{n}p_{i}\log _{2}p_{i}} ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Suppose that you possess a $D$-ary encoding $\Gamma$ for the source $S$ that does not satisfy Kraft's Inequality. Specifically, in this problem, we assume that our encoding satisfies $\sum_{i=1}^n D^{-l_i} = k+1 $ with $k>0$. What can you infer on the average code-word length $L(S,\Gamma)$? | This is an immediate consequence of the Kraft-McMillan inequality. Kraft's inequality states that given a sequence of strings { x i } i = 1 n {\displaystyle \{x_{i}\}_{i=1}^{n}} there exists a prefix code with codewords { σ i } i = 1 n {\displaystyle \{\sigma _{i}\}_{i=1}^{n}} where ∀ i , | σ i | = k i {\displaystyle \... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider a source $S$ with some distribution $P_S$ over the alphabet $\mathcal{A} = \{a, b, c, d, e, f\}$. Consider the following encoding $\Gamma$ over a code alphabet $\mathcal{D}$ of size $D$ with the following codeword lengths: egin{center} egin{tabular}{ |c|c|c|c|c|c|c| } \hline & $a$ & $b$ & $c$ & $... | Let each source symbol from the alphabet S = { s 1 , s 2 , … , s n } {\displaystyle S=\{\,s_{1},s_{2},\ldots ,s_{n}\,\}} be encoded into a uniquely decodable code over an alphabet of size r {\displaystyle r} with codeword lengths ℓ 1 , ℓ 2 , … , ℓ n . {\displaystyle \ell _{1},\ell _{2},\ldots ,\ell _{n}.} Then ∑ i = 1 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider a source $S$ with some distribution $P_S$ over the alphabet $\mathcal{A} = \{a, b, c, d, e, f\}$. Consider the following encoding $\Gamma$ over a code alphabet $\mathcal{D}$ of size $D$ with the following codeword lengths: egin{center} egin{tabular}{ |c|c|c|c|c|c|c| } \hline & $a$ & $b$ & $c$ & $... | The source coding theorem for symbol codes places an upper and a lower bound on the minimal possible expected length of codewords as a function of the entropy of the input word (which is viewed as a random variable) and of the size of the target alphabet. Note that, for data that exhibits more dependencies (whose sourc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider a source $S$ with some distribution $P_S$ over the alphabet $\mathcal{A} = \{a, b, c, d, e, f\}$. Consider the following encoding $\Gamma$ over a code alphabet $\mathcal{D}$ of size $D$ with the following codeword lengths: egin{center} egin{tabular}{ |c|c|c|c|c|c|c| } \hline & $a$ & $b$ & $c$ & $... | Let each source symbol from the alphabet S = { s 1 , s 2 , … , s n } {\displaystyle S=\{\,s_{1},s_{2},\ldots ,s_{n}\,\}} be encoded into a uniquely decodable code over an alphabet of size r {\displaystyle r} with codeword lengths ℓ 1 , ℓ 2 , … , ℓ n . {\displaystyle \ell _{1},\ell _{2},\ldots ,\ell _{n}.} Then ∑ i = 1 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider a source $S$ with some distribution $P_S$ over the alphabet $\mathcal{A} = \{a, b, c, d, e, f\}$. Consider the following encoding $\Gamma$ over a code alphabet $\mathcal{D}$ of size $D$ with the following codeword lengths: egin{center} egin{tabular}{ |c|c|c|c|c|c|c| } \hline & $a$ & $b$ & $c$ & $... | Let C {\displaystyle {\mathcal {C}}} be a ( n , k , d ) q {\displaystyle (n,k,d)_{q}} error-correcting code; in other words, C {\displaystyle {\mathcal {C}}} is a code of length n {\displaystyle n} , dimension k {\displaystyle k} and minimum distance d {\displaystyle d} over an alphabet Σ {\displaystyle \Sigma } of siz... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider a source $S$ with some distribution $P_S$ over the alphabet $\mathcal{A} = \{a, b, c, d, e, f\}$. Consider the following encoding $\Gamma$ over a code alphabet $\mathcal{D}$ of size $D$ with the following codeword lengths: egin{center} egin{tabular}{ |c|c|c|c|c|c|c| } \hline & $a$ & $b$ & $c$ & $... | Let each source symbol from the alphabet S = { s 1 , s 2 , … , s n } {\displaystyle S=\{\,s_{1},s_{2},\ldots ,s_{n}\,\}} be encoded into a uniquely decodable code over an alphabet of size r {\displaystyle r} with codeword lengths ℓ 1 , ℓ 2 , … , ℓ n . {\displaystyle \ell _{1},\ell _{2},\ldots ,\ell _{n}.} Then ∑ i = 1 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider a source $S$ with some distribution $P_S$ over the alphabet $\mathcal{A} = \{a, b, c, d, e, f\}$. Consider the following encoding $\Gamma$ over a code alphabet $\mathcal{D}$ of size $D$ with the following codeword lengths: egin{center} egin{tabular}{ |c|c|c|c|c|c|c| } \hline & $a$ & $b$ & $c$ & $... | There is a bijection γ from the Lyndon words in an ordered alphabet to a basis of the free Lie algebra on this alphabet defined as follows: If a word w has length 1 then γ ( w ) = w {\displaystyle \gamma (w)=w} (considered as a generator of the free Lie algebra). If w has length at least 2, then write w = u v {\display... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider a source $S$ with some distribution $P_S$ over the alphabet $\mathcal{A} = \{a, b, c, d, e, f\}$. Consider the following encoding $\Gamma$ over a code alphabet $\mathcal{D}$ of size $D$ with the following codeword lengths: egin{center} egin{tabular}{ |c|c|c|c|c|c|c| } \hline & $a$ & $b$ & $c$ & $... | Let each source symbol from the alphabet S = { s 1 , s 2 , … , s n } {\displaystyle S=\{\,s_{1},s_{2},\ldots ,s_{n}\,\}} be encoded into a uniquely decodable code over an alphabet of size r {\displaystyle r} with codeword lengths ℓ 1 , ℓ 2 , … , ℓ n . {\displaystyle \ell _{1},\ell _{2},\ldots ,\ell _{n}.} Then ∑ i = 1 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider a source $S$ with some distribution $P_S$ over the alphabet $\mathcal{A} = \{a, b, c, d, e, f\}$. Consider the following encoding $\Gamma$ over a code alphabet $\mathcal{D}$ of size $D$ with the following codeword lengths: egin{center} egin{tabular}{ |c|c|c|c|c|c|c| } \hline & $a$ & $b$ & $c$ & $... | For the example mentioned above, the encoding becomes: (1,1,2), ('B','A','C','D') This means that the first symbol B is of length 1, then the A of length 2, and remains of 3. Since the symbols are sorted by bit-length, we can efficiently reconstruct the codebook. A pseudo code describing the reconstruction is introduce... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider the following sequence of random variables $S_1,\ldots,S_n,\ldots$ Assume that the limit $H^\star(\mathcal{S})=k$ exists and is finite. Suppose that there exists $\hat{n}>0$ such that for all $i\geq \hat{n}$ one has that the marginal distributions of $S_{i+1}$ and $S_i$ satisfy $p_{S_{i+1}}=p_{S_i}$. Denote wi... | Infinitely divisible distributions appear in a broad generalization of the central limit theorem: the limit as n → +∞ of the sum Sn = Xn1 + … + Xnn of independent uniformly asymptotically negligible (u.a.n.) random variables within a triangular array X 11 X 21 X 22 X 31 X 32 X 33 ⋮ ⋮ ⋮ ⋱ {\displaystyle {\begin{array}{c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider the following sequence of random variables $S_1,\ldots,S_n,\ldots$ Assume that the limit $H^\star(\mathcal{S})=k$ exists and is finite. Suppose that there exists $\hat{n}>0$ such that for all $i\geq \hat{n}$ one has that the marginal distributions of $S_{i+1}$ and $S_i$ satisfy $p_{S_{i+1}}=p_{S_i}$. Denote wi... | In order that the probability distribution of a random variable X {\displaystyle X} be uniquely defined by its moments α k = E X k {\displaystyle \alpha _{k}=EX^{k}} it is sufficient, for example, that Carleman's condition be satisfied: A similar result even holds for moments of random vectors. The problem of moments s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $S_{0},S_{1},S_{2},\dots$ be an infinite sequence produced by a source $\mathcal{S}$. All $S_{n}$ take values in $\{0,1\}$, and $S_{n+1}$ depends only on $S_n$, that is, $p_{S_{n+1} | S_0, \dots, S_n}(s_{n+1} | s_0, \dots, s_n) = p_{S_{n+1} | S_n}(s_{n+1} | s_n)$. The probability $p_{S_{n+1}|S_{n}}$ is schemati... | The probability of obtaining any one particular random graph with m edges is p m ( 1 − p ) N − m {\displaystyle p^{m}(1-p)^{N-m}} with the notation N = ( n 2 ) {\displaystyle N={\tbinom {n}{2}}} .A closely related model, the Erdős–Rényi model denoted G(n,M), assigns equal probability to all graphs with exactly M edges.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $S_{0},S_{1},S_{2},\dots$ be an infinite sequence produced by a source $\mathcal{S}$. All $S_{n}$ take values in $\{0,1\}$, and $S_{n+1}$ depends only on $S_n$, that is, $p_{S_{n+1} | S_0, \dots, S_n}(s_{n+1} | s_0, \dots, s_n) = p_{S_{n+1} | S_n}(s_{n+1} | s_n)$. The probability $p_{S_{n+1}|S_{n}}$ is schemati... | Let S i {\displaystyle S_{i}\,} denote the event that the source sequence X 1 n ( i ) {\displaystyle X_{1}^{n}(i)} was generated at the source, so that P ( S i ) = P ( X 1 n ( i ) ) . {\displaystyle P(S_{i})=P(X_{1}^{n}(i))\,.} Then the probability of error can be broken down as P ( E ) = ∑ i P ( E ∣ S i ) P ( S i ) . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $S_{0},S_{1},S_{2},\dots$ be an infinite sequence produced by a source $\mathcal{S}$. All $S_{n}$ take values in $\{0,1\}$, and $S_{n+1}$ depends only on $S_n$, that is, $p_{S_{n+1} | S_0, \dots, S_n}(s_{n+1} | s_0, \dots, s_n) = p_{S_{n+1} | S_n}(s_{n+1} | s_n)$. The probability $p_{S_{n+1}|S_{n}}$ is schemati... | Let β i ( t ) = P ( Y t + 1 = y t + 1 , … , Y T = y T ∣ X t = i , θ ) {\displaystyle \beta _{i}(t)=P(Y_{t+1}=y_{t+1},\ldots ,Y_{T}=y_{T}\mid X_{t}=i,\theta )} that is the probability of the ending partial sequence y t + 1 , … , y T {\displaystyle y_{t+1},\ldots ,y_{T}} given starting state i {\displaystyle i} at time t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $S_{0},S_{1},S_{2},\dots$ be an infinite sequence produced by a source $\mathcal{S}$. All $S_{n}$ take values in $\{0,1\}$, and $S_{n+1}$ depends only on $S_n$, that is, $p_{S_{n+1} | S_0, \dots, S_n}(s_{n+1} | s_0, \dots, s_n) = p_{S_{n+1} | S_n}(s_{n+1} | s_n)$. The probability $p_{S_{n+1}|S_{n}}$ is schemati... | Its formula is obtained by first calculating the dominant eigenvalue λ {\displaystyle \lambda } and corresponding eigenvector ψ {\displaystyle \psi } of the adjacency matrix, i.e. the largest λ ∈ R {\displaystyle \lambda \in \mathbb {R} } with corresponding ψ ∈ R n {\displaystyle \psi \in \mathbb {R} ^{n}} such that ψ ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
A colleague challenges you to create a $(n-1,k,d_{min})$ code $\mathcal C'$ from a $(n,k,d_{min})$ code $\mathcal C$ as follows: given a generator matrix $G$ that generates $\mathcal C$, drop one column from $G$. Then, generate the new code with this truncated $k imes (n-1)$ generator matrix. The catch is that your co... | Let G be a generator matrix of C. We can always suppose that the first row of G is of the form r = (1, ..., 1, 0, ..., 0) with weight d. G = {\displaystyle G={\begin{bmatrix}1&\dots &1&0&\dots &0\\\ast &\ast &\ast &&G'&\\\end{bmatrix}}} The matrix G ′ {\displaystyle G'} generates a code C ′ {\displaystyle C'} , which ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
A colleague challenges you to create a $(n-1,k,d_{min})$ code $\mathcal C'$ from a $(n,k,d_{min})$ code $\mathcal C$ as follows: given a generator matrix $G$ that generates $\mathcal C$, drop one column from $G$. Then, generate the new code with this truncated $k imes (n-1)$ generator matrix. The catch is that your co... | The RM(2,3) code is generated by the set: { v 0 , v 1 , v 2 , v 3 , v 1 ∧ v 2 , v 1 ∧ v 3 , v 2 ∧ v 3 } {\displaystyle \{v_{0},v_{1},v_{2},v_{3},v_{1}\wedge v_{2},v_{1}\wedge v_{3},v_{2}\wedge v_{3}\}} or more explicitly by the rows of the matrix: ( 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 1 0 0 1 1 0 0 1 1 1 1 0 0 0 0 1 0 0 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
A binary prefix-free code $\Gamma$ is made of four codewords. The first three codewords have codeword lengths $\ell_1 = 2$, $\ell_2 = 3$ and $\ell_3 = 3$. What is the minimum possible length for the fourth codeword? | Given A set of symbols and their weights (usually proportional to probabilities). Find A prefix-free binary code (a set of codewords) with minimum expected codeword length (equivalently, a tree with minimum weighted path length from the root). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
A binary prefix-free code $\Gamma$ is made of four codewords. The first three codewords have codeword lengths $\ell_1 = 2$, $\ell_2 = 3$ and $\ell_3 = 3$. What is the minimum possible length for the fourth codeword? | To maintain the prefix-free property, B's codeword may not start 00, so the lexicographically first available word of length 3 is 010. Continuing like this, we get the following code: Alternatively, we can use the cumulative probability method. Note that although the codewords under the two methods are different, the w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $n \geq 2$ be a positive integer, and $M$ a uniformly distributed binary message of length $2n$. Let $P_K(M)$ denote the one-time pad encryption of $M$ with key $K$. Let $K_1$ be a uniformly distributed binary key length $n$. Let $K_2$ be the complement of $K_1$. Let $K_3$ be the reverse of $K_1$. Let $K_i||K_j$ de... | Shannon proved, using information theoretic considerations, that the one-time pad has a property he termed perfect secrecy; that is, the ciphertext C gives absolutely no additional information about the plaintext. This is because (intuitively), given a truly uniformly random key that is used only once, a ciphertext can... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $n \geq 2$ be a positive integer, and $M$ a uniformly distributed binary message of length $2n$. Let $P_K(M)$ denote the one-time pad encryption of $M$ with key $K$. Let $K_1$ be a uniformly distributed binary key length $n$. Let $K_2$ be the complement of $K_1$. Let $K_3$ be the reverse of $K_1$. Let $K_i||K_j$ de... | The encryption exponent e and λ ( N ) {\displaystyle \lambda (N)} also must be relatively prime so that there is a modular inverse. The factorization of N and the private key d are kept secret, so that only Bob can decrypt the message. We denote the private key pair as (d, N). The encryption of the message M is given b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $n \geq 2$ be a positive integer, and $M$ a uniformly distributed binary message of length $2n$. Let $P_K(M)$ denote the one-time pad encryption of $M$ with key $K$. Let $K_1$ be a uniformly distributed binary key length $n$. Let $K_2$ be the complement of $K_1$. Let $K_3$ be the reverse of $K_1$. Let $K_i||K_j$ de... | Shannon proved, using information theoretic considerations, that the one-time pad has a property he termed perfect secrecy; that is, the ciphertext C gives absolutely no additional information about the plaintext. This is because (intuitively), given a truly uniformly random key that is used only once, a ciphertext can... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $n \geq 2$ be a positive integer, and $M$ a uniformly distributed binary message of length $2n$. Let $P_K(M)$ denote the one-time pad encryption of $M$ with key $K$. Let $K_1$ be a uniformly distributed binary key length $n$. Let $K_2$ be the complement of $K_1$. Let $K_3$ be the reverse of $K_1$. Let $K_i||K_j$ de... | Another way of stating perfect secrecy is that for all messages m 1 , m 2 {\displaystyle m_{1},m_{2}} in message space M, and for all ciphers c in cipher space C, we have Pr k ⇐ K = Pr k ⇐ K {\displaystyle {\underset {k\Leftarrow \mathrm {K} }{\operatorname {Pr} }}={\underset {k\Leftarrow \mathrm {K} }{\operatorname ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider an RSA encryption where the $(p, q)$ are determined as $(53, 61)$. True or false: $(e,d) = (7, 223)$ are valid encoding/decoding exponents. | Thus any d satisfying d⋅e ≡ 1 (mod φ(n)) also satisfies d⋅e ≡ 1 (mod λ(n)). However, computing d modulo φ(n) will sometimes yield a result that is larger than necessary (i.e. d > λ(n)). Most of the implementations of RSA will accept exponents generated using either method (if they use the private exponent d at all, rat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider an RSA encryption where the $(p, q)$ are determined as $(53, 61)$. True or false: $(e,d) = (7, 223)$ are valid encoding/decoding exponents. | This RSA modulus is made public together with the encryption exponent e. N and e form the public key pair (e, N). By making this information public, anyone can encrypt messages to Bob. The decryption exponent d satisfies e d = 1 mod λ ( N ) {\displaystyle ed=1{\bmod {\lambda }}(N)} , where λ ( N ) {\displaystyle \lambd... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider an RSA encryption where the $(p, q)$ are determined as $(53, 61)$. True or false: $(e,d) = (319, 23)$ are valid encoding/decoding exponents. | Thus any d satisfying d⋅e ≡ 1 (mod φ(n)) also satisfies d⋅e ≡ 1 (mod λ(n)). However, computing d modulo φ(n) will sometimes yield a result that is larger than necessary (i.e. d > λ(n)). Most of the implementations of RSA will accept exponents generated using either method (if they use the private exponent d at all, rat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider an RSA encryption where the $(p, q)$ are determined as $(53, 61)$. True or false: $(e,d) = (319, 23)$ are valid encoding/decoding exponents. | This RSA modulus is made public together with the encryption exponent e. N and e form the public key pair (e, N). By making this information public, anyone can encrypt messages to Bob. The decryption exponent d satisfies e d = 1 mod λ ( N ) {\displaystyle ed=1{\bmod {\lambda }}(N)} , where λ ( N ) {\displaystyle \lambd... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $G$ be a set and $*$ a commutative operation on pairs of elements from $G.$ Suppose there exists an element $e\in G$ such that $a*e=e*a=a$ for all $a \in G.$ Also, suppose there exist elements $b,c,d \in G$ such that $b*c=d*c.$. True or false: $(G,*)$ is a group if and only if $b=d.$ | Nevertheless, it is a monomorphism in this category. This follows from the implication q ∘ h = 0 ⇒ h = 0, which we will now prove. If h: G → Q, where G is some divisible group, and q ∘ h = 0, then h(x) ∈ Z, ∀ x ∈ G. Now fix some x ∈ G. Without loss of generality, we may assume that h(x) ≥ 0 (otherwise, choose −x instea... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $G$ be a set and $*$ a commutative operation on pairs of elements from $G.$ Suppose there exists an element $e\in G$ such that $a*e=e*a=a$ for all $a \in G.$ Also, suppose there exist elements $b,c,d \in G$ such that $b*c=d*c.$. True or false: $(G,*)$ is a group if and only if $b=d.$ | The general theory of abstract algebra allows an "addition" operation to be any associative and commutative operation on a set. Basic algebraic structures with such an addition operation include commutative monoids and abelian groups. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $G$ be a set and $*$ a commutative operation on pairs of elements from $G.$ Suppose there exists an element $e\in G$ such that $a*e=e*a=a$ for all $a \in G.$ Also, suppose there exist elements $b,c,d \in G$ such that $b*c=d*c.$. True or false: If $b
ot=d,$ then $(G,*)$ cannot be a group. | Nevertheless, it is a monomorphism in this category. This follows from the implication q ∘ h = 0 ⇒ h = 0, which we will now prove. If h: G → Q, where G is some divisible group, and q ∘ h = 0, then h(x) ∈ Z, ∀ x ∈ G. Now fix some x ∈ G. Without loss of generality, we may assume that h(x) ≥ 0 (otherwise, choose −x instea... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $G$ be a set and $*$ a commutative operation on pairs of elements from $G.$ Suppose there exists an element $e\in G$ such that $a*e=e*a=a$ for all $a \in G.$ Also, suppose there exist elements $b,c,d \in G$ such that $b*c=d*c.$. True or false: If $b
ot=d,$ then $(G,*)$ cannot be a group. | The general theory of abstract algebra allows an "addition" operation to be any associative and commutative operation on a set. Basic algebraic structures with such an addition operation include commutative monoids and abelian groups. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $G_1, G_2, G_3$, be valid generator matrices of dimensions $\mathbb F^{k_i imes n_i}$, all over the same field $\mathbb F$. Recall that ``valid'' means that for all $i$, $k_i \leq n_i$ and $ ext{rank}(G_i) = k_i$. True or false: Assuming $k_1 = k_2 + k_3$, the matrix $egin{pmatrix} G_1 &\vline &egin{matrix} G_2... | Its generators are the matrices g k = ( 1 + 2 ( 2 + 2 ) α e i k π 4 ( 2 + 2 ) α e − i k π 4 1 + 2 ) , {\displaystyle g_{k}={\begin{pmatrix}1+{\sqrt {2}}&(2+{\sqrt {2}})\alpha e^{\tfrac {ik\pi }{4}}\\(2+{\sqrt {2}})\alpha e^{-{\tfrac {ik\pi }{4}}}&1+{\sqrt {2}}\end{pmatrix}},} where α = 2 − 1 {\displaystyle \alpha ={\sq... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $G_1, G_2, G_3$, be valid generator matrices of dimensions $\mathbb F^{k_i imes n_i}$, all over the same field $\mathbb F$. Recall that ``valid'' means that for all $i$, $k_i \leq n_i$ and $ ext{rank}(G_i) = k_i$. True or false: Assuming $k_1 = k_2 + k_3$, the matrix $egin{pmatrix} G_1 &\vline &egin{matrix} G_2... | {\displaystyle {\begin{pmatrix}a&0\\0&{\frac {1}{a}}\\\end{pmatrix}},a\in \mathbb {R} ^{\times };\quad {\begin{pmatrix}1&x\\0&1\\\end{pmatrix}},x\in \mathbb {R} ;\quad S={\begin{pmatrix}0&-1\\1&0\\\end{pmatrix}}.} One calculates the claim for these generators and obtains the claim for all g ∈ S L 2 ( R ) {\displaystyle... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $G$, be a valid generator matrix of dimensions $\mathbb F^{k imes n}$. Recall that ``valid'' means that $k \leq n$ and $ ext{rank}(G) = k$. Let $D_1 \in \mathbb F^{k imes k}$ and $D_2 \in \mathbb F^{n imes n}$ be diagonal matrices with non-zero diagonal elements. True or false: $D_1 \cdot G \cdot D_2$ is also a ... | Its generators are the matrices g k = ( 1 + 2 ( 2 + 2 ) α e i k π 4 ( 2 + 2 ) α e − i k π 4 1 + 2 ) , {\displaystyle g_{k}={\begin{pmatrix}1+{\sqrt {2}}&(2+{\sqrt {2}})\alpha e^{\tfrac {ik\pi }{4}}\\(2+{\sqrt {2}})\alpha e^{-{\tfrac {ik\pi }{4}}}&1+{\sqrt {2}}\end{pmatrix}},} where α = 2 − 1 {\displaystyle \alpha ={\sq... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $G$, be a valid generator matrix of dimensions $\mathbb F^{k imes n}$. Recall that ``valid'' means that $k \leq n$ and $ ext{rank}(G) = k$. Let $D_1 \in \mathbb F^{k imes k}$ and $D_2 \in \mathbb F^{n imes n}$ be diagonal matrices with non-zero diagonal elements. True or false: $D_1 \cdot G \cdot D_2$ is also a ... | {\displaystyle {\begin{pmatrix}a&0\\0&{\frac {1}{a}}\\\end{pmatrix}},a\in \mathbb {R} ^{\times };\quad {\begin{pmatrix}1&x\\0&1\\\end{pmatrix}},x\in \mathbb {R} ;\quad S={\begin{pmatrix}0&-1\\1&0\\\end{pmatrix}}.} One calculates the claim for these generators and obtains the claim for all g ∈ S L 2 ( R ) {\displaystyle... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $G_1, G_2$, be valid generator matrices of dimensions $\mathbb F^{k_i imes n_i}$, all over the same field $\mathbb F$. Recall that ``valid'' means that for all $i$, $k_i \leq n_i$ and $ ext{rank}(G_i) = k_i$. True or false: Assuming $k_1 = k_2$ and $n_1=n_2$, the matrix $G_{7} + G_{8}$ is also a valid generator m... | Its generators are the matrices g k = ( 1 + 2 ( 2 + 2 ) α e i k π 4 ( 2 + 2 ) α e − i k π 4 1 + 2 ) , {\displaystyle g_{k}={\begin{pmatrix}1+{\sqrt {2}}&(2+{\sqrt {2}})\alpha e^{\tfrac {ik\pi }{4}}\\(2+{\sqrt {2}})\alpha e^{-{\tfrac {ik\pi }{4}}}&1+{\sqrt {2}}\end{pmatrix}},} where α = 2 − 1 {\displaystyle \alpha ={\sq... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $G_1, G_2$, be valid generator matrices of dimensions $\mathbb F^{k_i imes n_i}$, all over the same field $\mathbb F$. Recall that ``valid'' means that for all $i$, $k_i \leq n_i$ and $ ext{rank}(G_i) = k_i$. True or false: Assuming $k_1 = k_2$ and $n_1=n_2$, the matrix $G_{7} + G_{8}$ is also a valid generator m... | {\displaystyle {\begin{pmatrix}a&0\\0&{\frac {1}{a}}\\\end{pmatrix}},a\in \mathbb {R} ^{\times };\quad {\begin{pmatrix}1&x\\0&1\\\end{pmatrix}},x\in \mathbb {R} ;\quad S={\begin{pmatrix}0&-1\\1&0\\\end{pmatrix}}.} One calculates the claim for these generators and obtains the claim for all g ∈ S L 2 ( R ) {\displaystyle... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $\mathcal C_1$ be a linear code over $\mathbb F_3^n$, and let $\mathcal C_2$ be a linear code over $\mathbb F_2^n$. True or false: $\mathcal C_1 \cap \mathcal C_2$ is necessarily a linear code over $\mathbb F_3^n$. | Let C ⊆ F 2 n {\displaystyle C\subseteq F_{2}^{n}} be a linear code such that C ⊥ {\displaystyle C^{\perp }} has distance greater than ℓ + 1 {\displaystyle \ell +1} . Then C {\displaystyle C} is an ℓ {\displaystyle \ell } -wise independent source. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $\mathcal C_1$ be a linear code over $\mathbb F_3^n$, and let $\mathcal C_2$ be a linear code over $\mathbb F_2^n$. True or false: $\mathcal C_1 \cap \mathcal C_2$ is necessarily a linear code over $\mathbb F_3^n$. | The construction thus gives the linear map: (cf. Lemma 1) C k G → H 2 k ( M ; C ) , f ↦ . {\displaystyle \mathbb {C} _{k}^{G}\to H^{2k}(M;\mathbb {C} ),\,f\mapsto \left.} In fact, one can check that the map thus obtained: C G → H ∗ ( M ; C ) {\displaystyle \mathbb {C} ^{G}\to H^{*}(M;\mathbb {C} )} is an algebra hom... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $$G= egin{pmatrix} 1 &1 &1 &0 &1 &0\ 0 &1 &1 &1 &0 &0\ 0 &1 &1 &0 &0 &0\ 0 &1 &1 &1 &0 &1 \end{pmatrix}$$ be the generator matrix of a $(6,4)$ linear code $\mathcal C$ over $\mathbb F_2$. True or false: $G$ admits a systematic form (i.e., it can be put into systematic form via elementary row operations). | Let G be a generator matrix of C. We can always suppose that the first row of G is of the form r = (1, ..., 1, 0, ..., 0) with weight d. G = {\displaystyle G={\begin{bmatrix}1&\dots &1&0&\dots &0\\\ast &\ast &\ast &&G'&\\\end{bmatrix}}} The matrix G ′ {\displaystyle G'} generates a code C ′ {\displaystyle C'} , which ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $$G= egin{pmatrix} 1 &1 &1 &0 &1 &0\ 0 &1 &1 &1 &0 &0\ 0 &1 &1 &0 &0 &0\ 0 &1 &1 &1 &0 &1 \end{pmatrix}$$ be the generator matrix of a $(6,4)$ linear code $\mathcal C$ over $\mathbb F_2$. True or false: $G$ admits a systematic form (i.e., it can be put into systematic form via elementary row operations). | Using the systematic construction for Hamming codes from above, the matrix A is apparent and the systematic form of G is written as G = ( 1 0 0 0 0 1 1 1 0 1 0 0 1 0 1 1 0 0 1 0 1 1 0 1 0 0 0 1 1 1 1 0 ) 4 , 8 . {\displaystyle \mathbf {G} =\left({\begin{array}{cccc|cccc}1&0&0&0&0&1&1&1\\0&1&0&0&1&0&1&1\\0&0&1&0&1&1&0&1... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $$G= egin{pmatrix} 1 &1 &1 &0 &1 &0\ 0 &1 &1 &1 &0 &0\ 0 &1 &1 &0 &0 &0\ 0 &1 &1 &1 &0 &1 \end{pmatrix}$$ be the generator matrix of a $(6,4)$ linear code $\mathcal C$ over $\mathbb F_2$. True or false: If one substitutes the last row of $G$ by $(1,0,0,1,1,1)$, the thereby obtained matrix generates the same code ... | Let G be a generator matrix of C. We can always suppose that the first row of G is of the form r = (1, ..., 1, 0, ..., 0) with weight d. G = {\displaystyle G={\begin{bmatrix}1&\dots &1&0&\dots &0\\\ast &\ast &\ast &&G'&\\\end{bmatrix}}} The matrix G ′ {\displaystyle G'} generates a code C ′ {\displaystyle C'} , which ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $$G= egin{pmatrix} 1 &1 &1 &0 &1 &0\ 0 &1 &1 &1 &0 &0\ 0 &1 &1 &0 &0 &0\ 0 &1 &1 &1 &0 &1 \end{pmatrix}$$ be the generator matrix of a $(6,4)$ linear code $\mathcal C$ over $\mathbb F_2$. True or false: If one substitutes the last row of $G$ by $(1,0,0,1,1,1)$, the thereby obtained matrix generates the same code ... | Let C ( x ) = c = ( c 0 , … , c 2 n − 1 ) {\displaystyle C(x)=c=(c_{0},\dots ,c_{2^{n}-1})} be the codeword in C {\displaystyle C} corresponding to message x {\displaystyle x} . Let G = ( ↑ ↑ ↑ g 0 g 1 … g 2 n − 1 ↓ ↓ ↓ ) {\displaystyle G={\begin{pmatrix}\uparrow &\uparrow &&\uparrow \\g_{0}&g_{1}&\dots &g_{2^{n}-1}\\\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $\mathbb F$ be a field of cardinality $q$ and let $0<k<n\leq q$ be unspecified integers. As seen in the lecture, we generate a $(n,k,d_{min})$ Reed-Solomon code with the following mapping: $$\mathbb F^k
ightarrow \mathbb F^n ~~,~~ \vec u \mapsto \vec c =(P_{\vec u}(a_1),P_{\vec u}(a_2),\ldots,P_{\vec u}(a_n))$$ fo... | Consider a ( n , k ) {\displaystyle (n,k)} Reed–Solomon code over the finite field F = G F ( q ) {\displaystyle \mathbb {F} =GF(q)} with evaluation set ( α 1 , α 2 , … , α n ) {\displaystyle (\alpha _{1},\alpha _{2},\ldots ,\alpha _{n})} and a positive integer r {\displaystyle r} , the Guruswami-Sudan List Decoder acce... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $\mathbb F$ be a field of cardinality $q$ and let $0<k<n\leq q$ be unspecified integers. As seen in the lecture, we generate a $(n,k,d_{min})$ Reed-Solomon code with the following mapping: $$\mathbb F^k
ightarrow \mathbb F^n ~~,~~ \vec u \mapsto \vec c =(P_{\vec u}(a_1),P_{\vec u}(a_2),\ldots,P_{\vec u}(a_n))$$ fo... | Using low-degree polynomials over a finite field F {\displaystyle \mathbb {F} } of size q {\displaystyle q} , it is possible to extend the definition of Reed–Muller codes to alphabets of size q {\displaystyle q} . Let m {\displaystyle m} and d {\displaystyle d} be positive integers, where m {\displaystyle m} should be ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider the source $S_1, S_2, \dots$ such that $S_1$ is uniformly distributed on $\mathbb{Z}/10\mathbb{Z}^*$, and for every $n\geq 1$, $S_{n+1}$ is distributed uniformly on $\mathbb{Z}/(S_n+1)\mathbb{Z}^*$. Let $H(\mathcal{S}) = \lim_{n o\infty} H(S_n)$. True or false: $H(\mathcal{S}) = 0$. | A sequence (s1, s2, s3, ...) of real numbers is said to be well-distributed on if for any subinterval of we have lim n → ∞ | { s k + 1 , … , s k + n } ∩ | n = d − c b − a {\displaystyle \lim _{n\to \infty }{\left|\{\,s_{k+1},\dots ,s_{k+n}\,\}\cap \right| \over n}={d-c \over b-a}} uniformly in k. Clearly every well... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Consider the source $S_1, S_2, \dots$ such that $S_1$ is uniformly distributed on $\mathbb{Z}/10\mathbb{Z}^*$, and for every $n\geq 1$, $S_{n+1}$ is distributed uniformly on $\mathbb{Z}/(S_n+1)\mathbb{Z}^*$. Let $H(\mathcal{S}) = \lim_{n o\infty} H(S_n)$. True or false: $H(\mathcal{S}) = 0$. | The multiplication theorem for the Hurwitz zeta function ζ ( s , a ) = ∑ n = 0 ∞ ( n + a ) − s {\displaystyle \zeta (s,a)=\sum _{n=0}^{\infty }(n+a)^{-s}} gives a distribution relation ∑ p = 0 q − 1 ζ ( s , a + p / q ) = q s ζ ( s , q a ) . {\displaystyle \sum _{p=0}^{q-1}\zeta (s,a+p/q)=q^{s}\,\zeta (s,qa)\ .} Hence f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $S_1$ be a random variable taking values in $\{a,b\}$ with probability $p_{S_1}(a) = rac{1}{4}$ and $p_{S_1}(b) = rac{3}{4}$. Let $S_2$ be a random variable, independent of $S_1$, taking values in $\{c,d\}$ with probability $p_{S_2}(c) = q$ and $p_{S_2}(d) = 1-q$, for some $q\in[0,1]$. Let $\Gamma_H$ be the binar... | Let Σ1, Σ2 denote two finite alphabets and let Σ∗1 and Σ∗2 denote the set of all finite words from those alphabets (respectively). Suppose that X is a random variable taking values in Σ1 and let f be a uniquely decodable code from Σ∗1 to Σ∗2 where |Σ2| = a. Let S denote the random variable given by the length of codewo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $S_1$ be a random variable taking values in $\{a,b\}$ with probability $p_{S_1}(a) = rac{1}{4}$ and $p_{S_1}(b) = rac{3}{4}$. Let $S_2$ be a random variable, independent of $S_1$, taking values in $\{c,d\}$ with probability $p_{S_2}(c) = q$ and $p_{S_2}(d) = 1-q$, for some $q\in[0,1]$. Let $\Gamma_H$ be the binar... | The average code length is L C ( X ) = ∑ x ∈ X p ( x ) L ( x ) = ∑ x ∈ X p ( x ) ( ⌈ log 2 1 p ( x ) ⌉ + 1 ) {\displaystyle LC(X)=\sum _{x\in X}p(x)L(x)=\sum _{x\in X}p(x)\left(\left\lceil \log _{2}{\frac {1}{p(x)}}\right\rceil +1\right)} . Thus for H(X), the entropy of the random variable X, H ( X ) + 1 ≤ L C ( X ) ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $S$ be a random variable taking values in $\{a,b,c,d,e\}$ with the following probabilities. $$egin{array}{|c|c|c|c|c|c|} \hline & a & b & c & d & e \ \hline p_S(\cdot) & 1/3 & 1/3 & 1/9 & 1/9 & 1/9\ \hline \end{array}$$ Let $\Gamma_D$ be the $D$-ary Huffman code for $S$. Let $L(S,\Gamma_D)$ be the average codeword... | The average code length is L C ( X ) = ∑ x ∈ X p ( x ) L ( x ) = ∑ x ∈ X p ( x ) ( ⌈ log 2 1 p ( x ) ⌉ + 1 ) {\displaystyle LC(X)=\sum _{x\in X}p(x)L(x)=\sum _{x\in X}p(x)\left(\left\lceil \log _{2}{\frac {1}{p(x)}}\right\rceil +1\right)} . Thus for H(X), the entropy of the random variable X, H ( X ) + 1 ≤ L C ( X ) ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Let $S$ be a random variable taking values in $\{a,b,c,d,e\}$ with the following probabilities. $$egin{array}{|c|c|c|c|c|c|} \hline & a & b & c & d & e \ \hline p_S(\cdot) & 1/3 & 1/3 & 1/9 & 1/9 & 1/9\ \hline \end{array}$$ Let $\Gamma_D$ be the $D$-ary Huffman code for $S$. Let $L(S,\Gamma_D)$ be the average codeword... | We give an example of the result of Huffman coding for a code with five characters and given weights. We will not verify that it minimizes L over all codes, but we will compute L and compare it to the Shannon entropy H of the given set of weights; the result is nearly optimal. For any code that is biunique, meaning tha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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