Formatting fixes

experiment/posttest.json

@@ -11,11 +11,11 @@
         "e": "None"
       },
       "explanations": {
-        "a": "For the given generator matrix, we have number of rows as $k=3$ and the blocklength (number of columns) as $n=6$. Further the three rows of $G$ are linearly independent. Thus, any parity check matrix $H$ should have $n-k=3$ linearly independent rows, and $6$ columns, and should satisfy the property $GH^T=\\boldsymbol{0}_{k\\times n-k}$. The matrix in this option has exactly all these properties. Hence it is a valid parity check matrix. To find the correct answer, look for a matrix which does <b>not have<b> at least one of these three properties.",
-        "b": "For the given generator matrix, we have number of rows as $k=3$ and the blocklength (number of columns) as $n=6$. Further the three rows of $G$ are linearly independent. Thus, any parity check matrix $H$ should have $n-k=3$ linearly independent rows, and $6$ columns, and should satisfy the property $GH^T=\\boldsymbol{0}_{k\\times n-k}$. The matrix in this option has exactly all these properties. Hence it is a valid parity check matrix. To find the correct answer, look for a matrix which does <b>not have<b> at least one of these three properties.", 
-        "c": "For the given generator matrix, we have number of rows as $k=3$ and the blocklength (number of columns) as $n=6$. Further the three rows of $G$ are linearly independent. Thus, any parity check matrix $H$ should have $n-k=3$ linearly independent rows, and $6$ columns, and should satisfy the property $GH^T=\\boldsymbol{0}_{k\\times n-k}$. The matrix in this option has exactly all these properties. Hence it is a valid parity check matrix. To find the correct answer, look for a matrix which does <b>not have<b> at least one of these three properties.",
-        "d": "For the given generator matrix, we have number of rows as $k=3$ and the blocklength (number of columns) as $n=6$. Further the three rows of $G$ are linearly independent. Thus, any parity check matrix $H$ should have $n-k=3$ linearly independent rows, and $6$ columns, and should satisfy the property $GH^T=\\boldsymbol{0}_{k\\times n-k}$. The matrix in this option has exactly all these properties. Hence it is a valid parity check matrix. To find the correct answer, look for a matrix which does <b>not have<b> at least one of these three properties.",
-        "e": "Yes, this is the right answer! All options are valid parity-check matrix of given generator matrix."
+        "a": "Incorrect. For the given generator matrix, we have number of rows as $k=3$ and the blocklength (number of columns) as $n=6$. Further the three rows of $G$ are linearly independent. Thus, any parity check matrix $H$ should have $n-k=3$ linearly independent rows, and $6$ columns, and should satisfy the property $GH^T=\\boldsymbol{0}_{k\\times n-k}$. The matrix in this option has exactly all these properties. Hence it is a valid parity check matrix. To find the correct answer, look for a matrix which does <b>not have<b> at least one of these three properties.",
+        "b": "Incorrect. For the given generator matrix, we have number of rows as $k=3$ and the blocklength (number of columns) as $n=6$. Further the three rows of $G$ are linearly independent. Thus, any parity check matrix $H$ should have $n-k=3$ linearly independent rows, and $6$ columns, and should satisfy the property $GH^T=\\boldsymbol{0}_{k\\times n-k}$. The matrix in this option has exactly all these properties. Hence it is a valid parity check matrix. To find the correct answer, look for a matrix which does <b>not have<b> at least one of these three properties.",
+        "c": "Incorrect. For the given generator matrix, we have number of rows as $k=3$ and the blocklength (number of columns) as $n=6$. Further the three rows of $G$ are linearly independent. Thus, any parity check matrix $H$ should have $n-k=3$ linearly independent rows, and $6$ columns, and should satisfy the property $GH^T=\\boldsymbol{0}_{k\\times n-k}$. The matrix in this option has exactly all these properties. Hence it is a valid parity check matrix. To find the correct answer, look for a matrix which does <b>not have<b> at least one of these three properties.",
+        "d": "Incorrect. For the given generator matrix, we have number of rows as $k=3$ and the blocklength (number of columns) as $n=6$. Further the three rows of $G$ are linearly independent. Thus, any parity check matrix $H$ should have $n-k=3$ linearly independent rows, and $6$ columns, and should satisfy the property $GH^T=\\boldsymbol{0}_{k\\times n-k}$. The matrix in this option has exactly all these properties. Hence it is a valid parity check matrix. To find the correct answer, look for a matrix which does <b>not have<b> at least one of these three properties.",
+        "e": "Correct. All options are valid parity-check matrix of given generator matrix."
       },
       "correctAnswer": "e",
       "difficulty": "intermediate"
@@ -29,10 +29,10 @@
         "d": "$\\left(0,0,0,0,0,0\\right)$"
       },
       "explanations": {
-        "a": "Correct option",
-        "b": "Valid codeword but not correct w.r.t $\\textbf{m}$. For instance first cooardinate of the codeword corresponding to message vector will be $(1\\times 1 + 0\\times 1 + 0\\times 1 + 1\\times 1)\\mod2 = 0$.",
-        "c": "Valid codeword but not correct w.r.t $\\textbf{m}$. For instance first cooardinate of the codeword corresponding to message vector will be $(1\\times 1 + 0\\times 1 + 0\\times 1 + 1\\times 1)\\mod2 = 0$.",
-        "d": "Valid codeword but not correct w.r.t $\\textbf{m}$. For instance third cooardinate of the codeword corresponding to message vector will be $(1\\times 1 + 0\\times 0 + 0\\times 0 + 1\\times 0)\\mod2 = 1$."
+        "a": "Correct.",
+        "b": "Incorrect. Valid codeword but not correct w.r.t $\\textbf{m}$. For instance first cooardinate of the codeword corresponding to message vector will be $(1\\times 1 + 0\\times 1 + 0\\times 1 + 1\\times 1)\\mod2 = 0$.",
+        "c": "Incorrect. Valid codeword but not correct w.r.t $\\textbf{m}$. For instance first cooardinate of the codeword corresponding to message vector will be $(1\\times 1 + 0\\times 1 + 0\\times 1 + 1\\times 1)\\mod2 = 0$.",
+        "d": "Incorrect. Valid codeword but not correct w.r.t $\\textbf{m}$. For instance third cooardinate of the codeword corresponding to message vector will be $(1\\times 1 + 0\\times 0 + 0\\times 0 + 1\\times 0)\\mod2 = 1$."
       },
       "correctAnswer": "a",
       "difficulty": "beginner"
@@ -46,10 +46,10 @@
         "d": "$\\left(0,1,1,1\\right)$"
       },
       "explanations": {
-        "a": "A valid codeword, $\\textbf{c}$, always satisfy $\\textbf{c}\\textbf{H} = \\textbf{0}$.",
-        "b": "Zero-vector is a valid codeword for any linear code. A valid codeword, $\\textbf{c}$, always satisfy $\\textbf{c}\\textbf{H} = \\textbf{0}$.",
-        "c": "A valid codeword, $\\textbf{c}$, always satisfy $\\textbf{c}\\textbf{H} = \\textbf{0}$.",
-        "d": "Correct Option! An invalid vector, $\\textbf{c}$, always satisfy $\\textbf{c}\\textbf{H} \\neq \\textbf{0}$."
+        "a": "Incorrect. A valid codeword, $\\textbf{c}$, always satisfy $\\textbf{c}\\textbf{H} = \\textbf{0}$.",
+        "b": "Incorrect. Zero-vector is a valid codeword for any linear code. A valid codeword, $\\textbf{c}$, always satisfy $\\textbf{c}\\textbf{H} = \\textbf{0}$.",
+        "c": "Incorrect. A valid codeword, $\\textbf{c}$, always satisfy $\\textbf{c}\\textbf{H} = \\textbf{0}$.",
+        "d": "Correct. An invalid vector, $\\textbf{c}$, always satisfy $\\textbf{c}\\textbf{H} \\neq \\textbf{0}$."
       },
       "correctAnswer": "d",
       "difficulty": "beginner"
@@ -63,10 +63,10 @@
         "d": "$G = \\begin{bmatrix} 1 & 0 & 1 & 0 \\\\ 1 & 1 & 1 & 0 \\end{bmatrix}$ and message transmitted be $\\left(1,1\\right)$"
       },
       "explanations": {
-        "a": "Valid Generator matrix but wrong message vector.",
-        "b": "Invalid Generator matrix corresponding to given code.",
-        "c": "Correct option!",
-        "d": "Invalid Generator matrix corresponding to given code."
+        "a": "Incorrect. Valid Generator matrix but wrong message vector.",
+        "b": "Incorrect. Invalid Generator matrix corresponding to given code.",
+        "c": "Correct.",
+        "d": "Incorrect. Invalid Generator matrix corresponding to given code."
       },
       "correctAnswer": "c",
       "difficulty": "beginner"
@@ -80,13 +80,13 @@
         "d": "If the message is $\\left(0,0\\right)$, the corresponding codeword is $\\left(1,0,1,0\\right)$, given $a=1$."
       },
       "explanations": {
-        "a": "For $\\left(1,0\\right)$ and $a=0$ codeword be $\\left(1,0,1,0\\right)$",
-        "b": "For $\\left(1,1\\right)$ and $a=1$ codeword be $\\left(1,0,0,1\\right)$",
-        "c": "Correct option",
-        "d": "For $\\left(0,0\\right)$ and $a=0$ or $1$ codeword be $\\left(0,0,0,0\\right)$"
+        "a": "Incorrect. For $\\left(1,0\\right)$ and $a=0$ codeword be $\\left(1,0,1,0\\right)$",
+        "b": "Incorrect. For $\\left(1,1\\right)$ and $a=1$ codeword be $\\left(1,0,0,1\\right)$",
+        "c": "Correct.",
+        "d": "Incorrect. For $\\left(0,0\\right)$ and $a=0$ or $1$ codeword be $\\left(0,0,0,0\\right)$"
       },
       "correctAnswer": "c",
       "difficulty": "beginner"
     }
   ]
-}
+}

experiment/pretest.json

@@ -10,10 +10,10 @@
         "d": "$\\left(0,0,0,1,0,0\\right)$"
       },
       "explanations": {
-        "a": "Please think about the modulo-2 sum of bits in each coordinate in the two vectors. For instance, the first position should be $(1+1)\\mod2 = 0$ and third position should be $(1+0)\\mod2 = 1$.",
-        "b": "Correct Answer!, the modulo-2 sum of each coordinate $((1+1)\\mod2,(0+0)\\mod2,(1+0)\\mod2,(0+1)\\mod2,(1+1)\\mod2,(0+1)\\mod2) = (0,0,1,1,0,1)$.",
-        "c": "Please think about the modulo-2 sum of bits in each coordinate in the two vectors. For instance, the first position should be $(1+1)\\mod2 = 0$ and third position should be $(0+1)\\mod2 = 1$.",
-        "d": "Please think about the modulo-2 sum of bits in each coordinate in the two vectors. For instance, the first position should be $(1+1)\\mod2 = 0$ and third position should be $(0+1)\\mod2 = 1$."
+        "a": "Incorrect. Please think about the modulo-2 sum of bits in each coordinate in the two vectors. For instance, the first position should be $(1+1)\\mod2 = 0$ and third position should be $(1+0)\\mod2 = 1$.",
+        "b": "Correct. The modulo-2 sum of each coordinate $((1+1)\\mod2,(0+0)\\mod2,(1+0)\\mod2,(0+1)\\mod2,(1+1)\\mod2,(0+1)\\mod2) = (0,0,1,1,0,1)$.",
+        "c": "Incorrect. Please think about the modulo-2 sum of bits in each coordinate in the two vectors. For instance, the first position should be $(1+1)\\mod2 = 0$ and third position should be $(0+1)\\mod2 = 1$.",
+        "d": "Incorrect. Please think about the modulo-2 sum of bits in each coordinate in the two vectors. For instance, the first position should be $(1+1)\\mod2 = 0$ and third position should be $(0+1)\\mod2 = 1$."
       },
       "correctAnswer": "b",
       "difficulty": "beginner"
@@ -27,10 +27,10 @@
         "d": "$\\begin{bmatrix} 0 & 1 & 1 \\\\ 1 & 0 & 1 \\\\ 1 & 1 & 1 \\end{bmatrix}$"
       },
       "explanations": {
-        "a": "Please think about the modulo-2 sum of bits in each coordinate in the two matrices. For instance, the $AB_{(1,1)}$ position should be $(1\\times 1 + 0\\times 0 + 1\\times 1)\\mod2 = 0$ and $AB_{(2,2)}$ position should be $(1\\times 1 + 1\\times 0 + 0\\times 0)\\mod2 = 1$.",
-        "b": "Correct Answer! the matrix we get after product is $\\begin{bmatrix} (1\\times 1 + 0\\times 0 + 1\\times 0)\\mod2 & (1\\times 1 + 0\\times 0 + 0\\times 0)\\mod2 & (1\\times 0 + 0\\times 1 + 1\\times 1)\\mod2 \\\\ (1\\times 1 + 1\\times 0 + 0\\times 1)\\mod2 & (1\\times 1 + 1\\times 0 + 1\\times 0)\\mod2 & (1\\times 0 + 1\\times 1 + 1\\times 1)\\mod2 \\\\ (0\\times 1 + 1\\times 0 + 1\\times 1)\\mod2 & (0\\times 1 + 1\\times 0 + 1\\times 0)\\mod2 & (0\\times 0 + 1\\times 1 + 1\\times 1)\\mod2 \\end{bmatrix} = \\begin{bmatrix} 0 & 1 & 1 \\\\ 1 & 1 & 1 \\\\ 1 & 0 & 1 \\end{bmatrix}$",
-        "c": "Please think about the modulo-2 sum of bits in each coordinate in the two matrices. For instance, the $AB_{(3,2)}$ position should be $(0\\times 1 + 1\\times 0 + 1\\times 0)\\mod2 = 0.$",
-        "d": "Please think about the modulo-2 sum of bits in each coordinate in the two matrices. For instance, the $AB_{(2,2)}$ position should be $(1\\times 1 + 1\\times 0 + 0\\times 0)\\mod2 = 1.$"
+        "a": "Incorrect. Please think about the modulo-2 sum of bits in each coordinate in the two matrices. For instance, the $AB_{(1,1)}$ position should be $(1\\times 1 + 0\\times 0 + 1\\times 1)\\mod2 = 0$ and $AB_{(2,2)}$ position should be $(1\\times 1 + 1\\times 0 + 0\\times 0)\\mod2 = 1$.",
+        "b": "Correct. The matrix we get after product is $\\begin{bmatrix} (1\\times 1 + 0\\times 0 + 1\\times 0)\\mod2 & (1\\times 1 + 0\\times 0 + 0\\times 0)\\mod2 & (1\\times 0 + 0\\times 1 + 1\\times 1)\\mod2 \\\\ (1\\times 1 + 1\\times 0 + 0\\times 1)\\mod2 & (1\\times 1 + 1\\times 0 + 1\\times 0)\\mod2 & (1\\times 0 + 1\\times 1 + 1\\times 1)\\mod2 \\\\ (0\\times 1 + 1\\times 0 + 1\\times 1)\\mod2 & (0\\times 1 + 1\\times 0 + 1\\times 0)\\mod2 & (0\\times 0 + 1\\times 1 + 1\\times 1)\\mod2 \\end{bmatrix} = \\begin{bmatrix} 0 & 1 & 1 \\\\ 1 & 1 & 1 \\\\ 1 & 0 & 1 \\end{bmatrix}$",
+        "c": "Incorrect. Please think about the modulo-2 sum of bits in each coordinate in the two matrices. For instance, the $AB_{(3,2)}$ position should be $(0\\times 1 + 1\\times 0 + 1\\times 0)\\mod2 = 0.$",
+        "d": "Incorrect. Please think about the modulo-2 sum of bits in each coordinate in the two matrices. For instance, the $AB_{(2,2)}$ position should be $(1\\times 1 + 1\\times 0 + 0\\times 0)\\mod2 = 1.$"
       },
       "correctAnswer": "b",
       "difficulty": "beginner"
@@ -44,10 +44,10 @@
         "d": "$\\left(0,1,0\\right)$"
       },
       "explanations": {
-        "a": "Please think about the modulo-2 sum of bits in each coordinate in the four vectors. For instance, the 3 position should be $(1+1+0+1) \\mod2 = 1$.",
-        "b": "Please think about the modulo-2 sum of bits in each coordinate in the four vectors. For instance, the 2 position should be $(0 + 0 + 0 + 1) \\mod2 = 1$.",
-        "c": "Correct Answer! sum of four tuple is like $((1 + 0 + 1 + 0) \\mod2,(0 + 0 + 0 + 1) \\mod2,(1 + 1 + 0 + 1) \\mod2) = (0,1,1)$",
-        "d": "Please think about the modulo-2 sum of bits in each coordinate in the four vectors. For instance, the 3 position should be $(1+1+0+1) \\mod2 = 1$."
+        "a": "Incorrect. Please think about the modulo-2 sum of bits in each coordinate in the four vectors. For instance, the 3 position should be $(1+1+0+1) \\mod2 = 1$.",
+        "b": "Incorrect. Please think about the modulo-2 sum of bits in each coordinate in the four vectors. For instance, the 2 position should be $(0 + 0 + 0 + 1) \\mod2 = 1$.",
+        "c": "Correct. The sum of four tuple is like $((1 + 0 + 1 + 0) \\mod2,(0 + 0 + 0 + 1) \\mod2,(1 + 1 + 0 + 1) \\mod2) = (0,1,1)$",
+        "d": "Incorrect. Please think about the modulo-2 sum of bits in each coordinate in the four vectors. For instance, the 3 position should be $(1+1+0+1) \\mod2 = 1$."
       },
       "correctAnswer": "c",
       "difficulty": "beginner"
@@ -61,10 +61,10 @@
         "d": "If we add the vector $\\left(1,0,1,0,1\\right)$ to $S$ and get a new set with $4$ vectors named $S'$, then the set $S'$ is a linear dependent set of vectors."
       },
       "explanations": {
-        "a": "Given option is linear dependent vector. Verify it be taking linear combination of $\\left(0,1,0,1,1\\right), \\text{ and } \\left(1,1,0,1,1\\right)$ will result in this option.",
-        "b": "Given option is linear dependent vector. Verify it be taking linear combination of $\\left(1,0,1,0,1\\right), \\text{ and } \\left(1,1,0,1,1\\right)$ will result in this option.",
-        "c": "Given option is linear independent vector. Verify it be taking any linear combination will not result this given vector.",
-        "d": "Correct Answer! Given option is linear dependent vector. The option is an element of the vecotr  S` thus it's linear dependent vector $\\left(1,1,0,1,1\\right)$."
+        "a": "Incorrect. Given option is linear dependent vector. Verify it be taking linear combination of $\\left(0,1,0,1,1\\right), \\text{ and } \\left(1,1,0,1,1\\right)$ will result in this option.",
+        "b": "Incorrect. Given option is linear dependent vector. Verify it be taking linear combination of $\\left(1,0,1,0,1\\right), \\text{ and } \\left(1,1,0,1,1\\right)$ will result in this option.",
+        "c": "Incorrect. Given option is linear independent vector. Verify it be taking any linear combination will not result this given vector.",
+        "d": "Correct. Given option is linear dependent vector. The option is an element of the vecotr  S` thus it's linear dependent vector $\\left(1,1,0,1,1\\right)$."
       },
       "correctAnswer": "d",
       "difficulty": "intermediate"
@@ -78,10 +78,10 @@
         "d": "5"
       },
       "explanations": {
-        "a": "Correct Answer! Row analysis: All 4 rows are linearly independent. Column analysis: Since second column is all zero thus rank is less than equal to 4. Rest 4 columns are linear independent.",
-        "b": "Please think about including 1 more linear independent rows or columns.",
-        "c": "Please think about including 2 more linear independent rows or columns.",
-        "d": "Number of rows is 4 then surely $rank(R) \\leq 4$"
+        "a": "Correct. Row analysis: All 4 rows are linearly independent. Column analysis: Since second column is all zero thus rank is less than equal to 4. Rest 4 columns are linear independent.",
+        "b": "Incorrect. Please think about including 1 more linear independent rows or columns.",
+        "c": "Incorrect. Please think about including 2 more linear independent rows or columns.",
+        "d": "Incorrect. Number of rows is 4 then surely $rank(R) \\leq 4$"
       },
       "correctAnswer": "a",
       "difficulty": "intermediate"