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Question:

अम्लों एवं क्षारकों, ऐल्कोहॉलों, सवंर्धनों और अन्य जैविक पदार्थों के लिए अपनाए जाने वाले निपटान का अनुसरण करने हेतु सुझावों को लिखिए।

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Question:

प्रयोगशाला कार्य की दो दिन पूर्व की तैयारी के लिए शैक्षिक कर्मी द्वारा प्रयोगशाला कर्मी को दी जाने वाली विस्तृत सूची को लिखिए

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Question:

कार्य-स्थल में मेमोरेन्डा का उद्देश्य क्या होता है? संप्रेषण के तरीकों में मेमोरेन्डम शीर्षकों की तुलना पत्रों के साथ कीजिए।

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Question:

प्रयोगशाला में भंडार कक्ष की रूपरेखा तैयार करते समय कौन-कौन सी महत्वपूर्ण बातों का ध्यान रखना चाहिए? मंडार कक्ष के उपयुक्त स्थान के लिए आवश्यक पर्यावरणीय कारकों को सूचीबद्ध कीजिए।

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Question:

प्रयोगशाला में पदार्थों और अन्य वस्तुओं के भंडारण के विभिन्न तरीके कौन-से होते हैं? प्रयोगशाला में विभिन्न प्रकार की वस्तुओं के भंडारण के समय ली जाने वाली सावधानियों को बताइए।

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Question:

विज्ञान प्रयोगशाला में तैयारी कक्ष का क्या महत्व होता है? तैयारी कक्ष के विभिन्न घटकों को सूचीबद्ध कीजिए।

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Question:

प्रयोगशाला में बेचिंग क्या होती हैं? बेचिंग व्यवस्थाओं के प्रकारों का वर्णन कीजिए।

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Question:

विज्ञान अन्वेषण का क्या अर्थ होता है? विज्ञान अन्वेषण के मुख्य तत्वों को सूचीबद्ध कीजिए।

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Question:

Find the convolutional code for the message 11011. The convolutional encoder is given in

Fig. 1.

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Question:

Draw the Tanner graph of the code C with parity check matrix

$\begin{bmatrix}1&0&0&0&1&0&0&1&0&1\\0&1&0&0&1&1&0&0&1&0\\0&0&1&0&0&1&1&0&0&1\\0&0&0&1&0&0&1&1&1&0\end{bmatrix}$

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Question:

Find the generating idempotents of duadic codes of length n = 23 over $\mathbb{F}_{3}$ .

(Hint: Mimic example 6.1.7.)

$C=\{0000,1113,2222,3331,0202,1313,2020,3131,0022,1131,2200,3313,0220,1333,2002,3111\}$

be the Z₄-linear code. Find the Gray image of C.

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Question:

Let C be the [5, 2] ternary code generated by

$G=\begin{pmatrix}1&1&0&0&1\\0&1&0&1&1\end{pmatrix}$

Find the weight enumerator W_C(x, y) of C.

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Question:

Let C be the ternary [8,3] narrow-sense BCH code of designed distance δ = 5, which has defining set T = {1,2,3,4,6}. Use the primitive root 8th root of unity you chose in 4a) to avoid recomputing the the table of powers. If

g(x) = x⁵- x⁴ + x³ + x² - 1

Image ignouassignments-ignouacademy-com--p-your-87418

Figure 1: Encoder for convolutional code.

is the generator polynomial of C and

y(x) = x⁷ - x⁶- x⁴ - x³

is the received word, find the transmitted codeword.

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Question:

Over $\mathbb{F}_{2}$ , (1+x) | (xⁿ - 1). Let C be the binary cyclic code (1+x) of length n. Let C₁ be any binary cyclic code of length n with generator polynomial g₁(x).

i) What is the dimension of ?

ii) Let w be subspace of $F_{2}^{n}$ containing all the vectors of even weight. Prove that W has dimension n - 1. (Hint: Consider the map w: $F_{2}^{n}$ → $F_{2}$ given by

$w((a_1,a_2,\dots,a_n))=a_1+a_2+\dots+a_n.$

Prove that & is the vector space of all vectors in $F_{2}^{n}$ with even weight.

iv) If C₁ has only even weight codewords, what is the relationship between (1+x) and g₁(x)?

v) If C₁ has some odd weight codewords, what is the relationship between 1 + x and g₁(x)?

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Question:

Let C₁ and C₂ be cyclic codes over $\mathbb{F}_{q}$ , with generator polynomials g₁ (x) and g₂(x), respectively. Prove that C₁ $\subseteq$ C₂ if and only if g₂(x) | g¹(x).

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Question:

Let a be a root of x² + 1 = 0 in $\mathbb{F}_{9}$ .

a) Check whether α is a primitive element of $\mathbb{F}_{9}$ . If it is not a primitive element in $\mathbb{F}_{9}$ find a primitive element γ in $\mathbb{F}_{9}$ in terms of a.

b) Make a table similiar to Table 5.1 on page 184 for $\mathbb{F}_{9}$ with the primitive element y

c) Factorise x⁸ over $\mathbb{F}_{3}$ .

d) Find all the possible generator polynomials of a [8, 6] cyclic code.

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Question:

The aim of this exercise is to show that every binary repetition code of odd length is perfect.

i) Find the value of t and d for a perfect code of length 2m + 1, m ∈ $\mathbb{N}$ .

ii) Show that

$\sum_{i=0}^{m}\binom{2m+1}{i}=2^{2m}$

(Hint: Start with the relation

$2^{2m+1}=\sum_{i=0}^{2m+1}\binom{2m+1}{i})$

iii) Deduce that every repetiition code of odd length is perfect.

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Question:

Show that, if x ∈ $\mathbb{F}_2^n$ ,

wt (x) = x.x (mod 3)

Deduce that, if C is a ternary self orthogonal code, the weight of each codeword is divisible by 3.

(Hint: Observe that x2 = 1 for all x ≠0 € $\mathbb{F}_{3}$ )

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Question:

Let C be a binary code with a generator matrix each of whose rows has even weight. Show that, every codeword of C has even weight.

(Hint: Why is it enough to prove that sum of vectors of even weight in $\mathbb{F}_2^n$ is a vector of even weight?)

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Question:

If $x,y\in\mathbb{F}_2^n$ , show that

wt(x+y)=wt(x)+wt (y) -2wt (xy)

where $x\cap y$ is the vector in $\mathbb{F}_2^n$ which has 1s precisely at those positions where x and y have 1s.

(Hint: Let x = (x₁,x₂,...,x_n) and y = (y₁, y₂,..., y_n). Suppose

$n_1=|\{i\mid x_i=y_i=1\}|,\quad n_2=|\{i\mid x_i=1,y_i=0\}|,\quad n_3=|\{i\mid x_i=0,y_i=1\}|$

Observe that wt (x) = n₁ + n₂ and wt (y) = n₁+n₃.)

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