 9.2.1: Let f be a homomorphism from a group 3G, # 4 to a group 3H, +4. Pro...
 9.2.2: Let f be a homorphism from the group 3Z, +4 to the group 3Z, +4 giv...
 9.2.3: The function f defined by f(x) = x # 8 2 is a homomorphism from 3Z,...
 9.2.4: The function f defined by f(x) = x # 8 4 is a homomorphism from 3Z1...
 9.2.5: A function f: Z Z S Z is defined by f(x, y) = x + y. a. Prove that ...
 9.2.6: Let F be the set of all functions f: R S R. For f, g [ F, let f + g...
 9.2.7: Let S = 50, 4, 86. Then 3S, +12 4 is a subgroup of 3Z12, +12 4. Fin...
 9.2.8: Let S = 5i,(2, 3)6. Then 3S, +4 is a subgroup of the symmetric grou...
 9.2.9: Consider the canonical paritycheck matrix h = F 1 1 1 0 1 1 1 0 1 ...
 9.2.10: Consider the canonical paritycheck matrix h = I 1 1 0 1 0 1 1 1 0 ...
 9.2.11: Give an example of a canonical paritycheck matrix that will genera...
 9.2.12: Give a canonical paritycheck matrix for a singleerror correcting ...
 9.2.13: Let H be an n r binary matrix mapping :Zn 2, +2; to :Zr 2, +2; by t...
 9.2.14: Which of the following are perfect codes? Which are singleerror co...
 9.2.15: Complete the coset leader/syndrome table of Example 28.
 9.2.16: Use your table from Exercise 15 and the matrix H from Example 28 to...
 9.2.17: For Exercises 17 and 18, use a canonical paritycheck matrix for th...
 9.2.18: For Exercises 17 and 18, use a canonical paritycheck matrix for th...
Solutions for Chapter 9.2: Coding Theory
Full solutions for Mathematical Structures for Computer Science  7th Edition
ISBN: 9781429215107
Solutions for Chapter 9.2: Coding Theory
Get Full SolutionsSince 18 problems in chapter 9.2: Coding Theory have been answered, more than 21560 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 9.2: Coding Theory includes 18 full stepbystep solutions. This textbook survival guide was created for the textbook: Mathematical Structures for Computer Science, edition: 7. Mathematical Structures for Computer Science was written by and is associated to the ISBN: 9781429215107.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Column space C (A) =
space of all combinations of the columns of A.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(DÂ» O.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.