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computer science
systems analysis design

Questions and Answers of Systems Analysis Design

Apply the algorithm to convert the following to nearly standard form 1 1 1 1 1 0 0 1 1 G = 1 0 1 0 0 1 1 0 1 0 1 01 over Z2. 1 1 0 0 0 1 1 1 1 2 2 0 0 0 0 0 0 G = 1 1 1 2 1 1 1 2 over Z3. 0 1 2 0 1
A ternary code hasDecode the received words lxyz12 and xyz21O. 2 1 2 1 10 H = 1 1 2 1 01 0 10 200
Why can a linear code of length 10 and dimension 6 never uniquely decode words with 5 erasures?
Two linear codes over Z5 have parity check matricesWhich code is better with respect to 2-erasure decodability? 1 3 124 2 4 1 1 2 2] and [3 3 2 1 0 4 4 2 1 3 2
Do a similar analysis for and r = x4423. 3 H = [ {} 1 2 3 4 10 4 0 1 ]
Show that the binary code withwill determine the transmitted word uniquely, on the assumption of at most one error, if r = 10110x. Demonstrate this result directly by using the list of codewords. H =
Decode the received ternary words lx20yl, 21xyll for a code with 2 1 2 1 10 H=112 10 1 0 1 0 2 0 0
Use the method of proving Theorem 6.1 to find a member of Ham(2, 5).
Show that the suggested column selection method does give a p.c. matrix of a member of Ham(r, q).
Show that any member of Ham(r, q) must have a p.c. matrix containing columns x, y, z whose first two members are aO, Ob, cc respectively, and the rest zero, for some non-zero a, b, c.
For C E Ham(5, 2) with the suggested column ordering decode the received word r with 1 in the first four positions and zeros elsewhere.
Find the 'convenient decoding' form for a p.c. matrix of an [8, 6, 3J Hamming code C, and use it to decode 12312300.
Show thatgenerates a binary Hamming code. 1 1 1 1 1 1 1 1 0 0 0 1 0 1 G = 1 1 0 0 0 10 0 1 1 0 0 0 1
Calculate the error-correcting capability of the dual of any code C in Harn (r,q).
Show that no linear equidistant ternary code C which is 2-error correcting and has dimension 4, length 10 can exist.[Hint: Imagine the code to be written out as an 81 x 10 array and, using Theorem
Show that all binary simplex codes are optimal.
Find upper and lower bounds for the size of the best binary linear 2-error correcting code of length 12.
In the construction used in our proof of Theorem 6.9, show that k is the dimension of the code constructed.
Show that any binary code with d = n is equivalent to a repetition code.
Show that perfect binary repetition codes must have odd length and that there are no perfect non-binary repetition codes with n > l.
Our definition of Res(C, c) seems to depend on G as weIl as onc. Show that this is not the case.
Let w(c) = w(c/) for distinct codewordsc, c' of C. Show that in generalRes( C, c) :I Res( C, c'), and indeed these two residual codes need not be equivalent, nor even have the same dimension.
Why must ac with the properties specified in Definition 6.3 exist?
For each real number x prove that =
With generator matrices chosen as in the preamble to Theorem 6.11 show thatwhere Res2 C means Res(ResC)··· etc. d(ResC) > d 22
Do a similar analysis comparing the performances of the Griesmer, Hamming and Plot kin bounds in upper bounding the size of [20, k, 11] binary codes.
Show that all binary simplex codes with dimension ~ 3 satisfy the conditions of Theorem 6.14 and are therefore self-orthogonal.
Explain why the previous two theorems do not generalize to the fieldsZp with p > 3.
Give an example to show that Theorem 6.18 does not extend to Zp withp> 3.
Find a self-dual Hamming code.
Find a binary [10, 5] self-dual code.
Show that if a binary self-orthogonal code C has a generator matrix G in which every row has a weight which is a multiple of 4 then every codeword weight is a multiple of 4. [ Rint : use the formula,
Check that H does indeed produce the codewords listed, and that the listings of C and D have the properties claimed.
Show that E f has 212 codewords.
Derive a formula for w(x + y + z) from the formula for w(x + y).
Show that for each a, b in Cf, a b has even weight.
Show that the Hamming code withis not cyclic. 1 0 0 1 1 10 H= 0 1 0 1 1 0 1 0 0 1 1 0 1 1
Prove the se co nd equality in our statement of the MacWilliams identity.
Show that there is no-error correcting code which is the dual of a repetition code.
Use the MacWilliams identity to prove that the dual of the binary repetition code of length n consists of all the even weight words of Z'2.Establish the result by an alternative method.
Verify from this expression that Al = A2 = 0, so that Harn (r, q) is 1-error correcting.
Explain how these expressions are obtained.
Use this recurrence relation to evaluate A3 ,A4 and A5 for Ham(4, 2).
Show that for Ram(3, 2) with p = 0.01 the actual probability of a decoding error is much sm aller than the upper bound just derived.
Let C be any binary code and Cf its extension by adding on overall parity check. How are their weight distributions Ai and A~ related?
A linear code Cover Z5 has generator matrixBy considering the weights of the codewords T2 and Tl + AT2, A =0,1,2,3,4, find the weight distribution of C.[Rint : consider multiples of the codewords
I,g E F[X]. What is the relation between deg(J) , deg(g) , deg(J + g),deg(f - g) and deg(fg)?
Find q and r when 3x6 + 2x5 + 4x3 + 2x + 2 in Z5 [Xl is divided by(a) 4x4 +x3 +x2 +3x+1,(b) 2x6 + x 4 + 3x2 ,(c) 3x+4.
Use long division to find q and r if(a) x 6 + x 3 + x 2 + X = q(x)(x4 + x 2 + X + 1) + r(x) in Z2[X],(b) x 6 + 2x5 + 2x + 1 = q(x)(2x3 + x 2 + X + 1) + r(x) in Z3[X].
Show that 1 +xlg(x) in Z2[X] if and only if g(x) has an even number of non-zero terms.
Show that, for 1 E Zp[X], for each ß E Zp\{O}, x - o:ll(x) if and only if ß(x - o:)ll(x).
Use the result of Exercise 5 to factorize each of the following into a constant and monie irreducibles.(a) 3x2 + 4x + 3 in Z5[X],(b) 2x3 + 2x2 + X + 2 in Z3[X],(c) X 3 + 4x2 + X + 1 in
Determine whether the quintie polynomial f(x) = (x + l)(x + 2)2(x +3)2 + 4 in Zs[X] is irreducible, and if not faetorize it eompletely.
Apply a polynomial version of Euclid's algorithm to find gcd(J, g) wheref(x) = x 12 + x4 + x3 + x 2 + X + 1, g(x) = x 8 + 2x6 + xS + x2 + 2x + 2, both in Z3[X],
Find the quadratic polynomial congruent to x 7 + 2x6 + 2x4 + x3 + x 2 + 2 mod x 3 + 2x(a) in Z3[X],(b) in Zs[X].
For each of the following cases determine whether (A, +, x) is(a) a ring,(b) a ring with unity,(e) a commutative ring.In (i), (ii) and (iii) + and x are ordinary matrix addition and
Let m(x) be a generator of the principal ideal I of F[X]. Show that for each a E F\ {O}, am( x) is also a generator of I. Show further that ifm(x) is a generator of minimal degree, then there
Show that F[X, Y], the set of polynomials in two variables with coefficients in a field F, is a ring, and that the subset I = {xs(x, y) +yt(x, y) :s E F[X, Y], t E F[X, Y]} is an ideal, but
Work out (2x3 + X + 1)(x4 + x 2 + 2x + 2) in Z3[X]/X3 + 2x2 + 1.
Prove that Zp[X]/ f(x) is a ring and that all its ideals are principal ideals.
Let R = Z2[X]/X3 - 1. Show that (1 + x) = (1 + x2).
Show that the rows of the matrix G whose form is specified in Theorem 8.2 are independent
Which of the following codes are (a) cyclic, (b) linearly equivalent to a cyclic code?(i) {OOOO, 1100, 0110, 0011, 1001} over Z2(ii) {OOOO, 1122, 2211} over Z3(iii) the q-a:ry repetition code over
(a) x n - 1 = (x - l)q(x) in Z2[X], What is q(x)?(b) Let 9 be the generator of the cyclic binary code C of length n. Show that if x - 1Ig(x) then all codewords have even weight.(c) Show
Is x 3 + 2x2 + 2 the generator of a cyclic ternary code of length 8 overZ3?
Find all the binary cyclic codes of length 21 having dimension 9
What is the generator polynomial and dimension of the smallest ternary code containing 112110? What is its minimum distance?
Use the method of this section to find a nearly standard G for the cyclic binary [7, 4] code generated by 1 + x 2 + x 3 .
Given that x6 + x 5 + x 4 + x 3 + 11x15 - lover Z2, find the codewordmG by the result of Theorem 8.8 where m = 010010001.
With the code of Exercise 11 find the syndrome of the received word 010011000111010.
Following on from Exercises 11 and 12, find the syndromes of all the cyclic shifts of the word 010011000111010.
Verify that over Z3 g(x) = x5 + x4 + 2x3 + x 2 + 2 is a divisor of xl! - 1.Let C be the cyclic ternary [11, 6] code (g(x)).Given that d( C) = 5, use error trapping to decode the received word
Write X 6p(x- 1) as a polynomial wherep( x) = 2x6 + 3x5 + x 3 + 4.
Find the generator polynomial for the cyclic member of Ham{3, 2) whose p.c. matrix is given in section 6.7.
Show that the cyclic code C is self orthogonal if and only if h(x)lg(x).Hence find a self orthogonal binary cyclic code with length 15.
Find a binary [15, 10] cyclic code which is all single, all double adjacent error correcting.
Let g be a k x n matrix whose first row is gOgl'" gn-kOO .. ·0 whereg(x) = go + glX + ... + gn_kxn-k is a polynomial of degree n - k withgo =J O. The remaining k - 1 rows are the first
There is a difficulty with this argument : there is nothing to preventj = k, but in this case our cancelling pairs ij,kl;kl,jl are not distinct, so the conclusion that w2 - w has to be a
Using the (unproved) uniqueness remarks in Chapter 6 concerning the Golay codes, prove the corollary to Theorem 8.14.
Show that if C, D are linear codes over Z3 with D ~ C and dim( C) = 1+ dim(D), then C is the union of three cosets of D.
Prove part (i) of Theorem 9.1.
Complete the proof of (ii) by showing that if u, u' E Cl and v, v' E C2 then the words ulu+v and u'lu' +v' of Cl * C2 cannot be equal, unless of course u = u' and v = v'.
Let Cl and C2 be binary linear codes of length n with generator matrices GI, G2 , respectively. Show that a generator matrix for Cl * C2 iswhere 0 represents the all-zero matrix of appropriate size.
Let C be a code of the family Ham (3,2), and D = C * C.(a) Find the dimension and minimum distance of D.(b) Is D a perfect code?(c) Give two reasons why, whichever Slepian array is chosen for D, not
List the code words of RM(2, 3) using our initial definition of RM(r, m).
Show that for all r, m, RM(r, m) is linear and has length 2m .
If G(r, m) is a generator matrix for RM(r, m) show that G(m, m) may be taken to be• and G(O, m) = [111 ... 1] (of length 2m ), and find a recursive definition of G(r, m) for 0 G(m-1,m)
By copying the technique for proving Theorem 9.2 and making use of Theorem 9.1 (part (iii)), prove Theorem 9.3.
Using Exercise 7 construct generator matrices for RM(2, 3) and RM(2, 4).
Show that there are Reed-Muller codes which are at least 7-error correcting and contain over 1000 codewords. Find one of shortest length.
Check that you can follow the derivation of the EB form of xy z + xy z by saying which of the rules 1-7 is being used at each step.
Derive the EB form of the Boolean function, xyz + xyz + xyz + xyz.
If C is any binary linear code, find a simple description of the code C *C.
Show from its definition that BF(r, m) is linear.
What is BF(r, m) when r ~ m?
Show that Theorems 9.2 and 9.4 have particularly easy proofs if the Boolean function description of RM(r, m) is used.
Carry out the basis step in the proof of Theorem 9.8.
Show that dim[RM(m - r -1, m)] = dim[RM(r, m).l], thus completing the proof of Theorem 9.8.
5. Robert has amassed a rather large set of digital video files on his computer. Many of them he imported himself from his digital camcorder, and others he (legally) downloaded from the Internet.
1. Teresa complains that her Windows XP notebook turns itself off without any warning. What should she adjust?

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