1. Lecture 19 The Vector Space Znp (Section 5.1)
Theory of Information Lecture 13
Theory of
Information
Lecture 19 The Vector Space Znp (Section 5.1)

2. About Linear Codes
Linear codes --- most important and widely studied types of codes.
Advantages of linear codes over other types of codes:
Easier to describe
Nearest neighbor decoding easier to implement
Encoding and decoding messages easier
Code alphabet is a field, so code symbols can be added and multiplied
Etc.
Our attention will focus on the fields Zp, where p is prime.
Remember that Zp={0,1,…,p-1}, and
Znp is the set of all strings over Zp of length n.
Remember everything else from Lecture 3.

3. Addition and Scalar Multiplication in Znp
Assume u=u1u2…unZnp, v=v1v2…vnZnp, and Zp.
Addition:
u+v = u1u2…un + v1v2…vn = (u1+v1)(u2+v2)…(un+vn)
Scalar multiplication:
u =   (u1u2…un) = (u1)(u2)…(un)
Example In Z43, =
2  (1212) =
Theorem If p is a prime number then Znp together with the above two
operations is a vector space over Zp.

4. Vector Spaces
Definition Let F be a field. A vector space over F is a nonempty set V together with
two functions: + (addition) of type VVV and  (scalar multiplication) of type
FVV such that these two operations satisfy the following conditions:
Associativity: For all x,y,zV, x+(y+z)=(x+y)+z
Commutativity: For all x,yV, x+y=y+x
Zero vector property: There exists an element 0V, called the zero vector, such that,
for all xV, 0+x=x+0=x
Property of negatives: For any xV, there exists another element of V, denoted by –x
and called the negative of x, such that (-x)+x=0
Scalar multiplication property: For all x,yV and ,F,
(x+y) = x+y
(+)x = x+x
()x = (x)
1x = x
The elements of V are called vectors, and the elements of F are called scalars.
F is called the base field of V.

5. Subspaces of Znp
Definition A nonempty subset S of Znp is called a subspace of Znp if the set S,
together with the operations of addition and scalar multiplication inherited from Znp,
is itself a vector space.
Theorem A nonempty subset S of Znp is a subspace of Znp if and only if it is
closed under addition and scalar multiplication.
I.e., for all vectors x,yS and every scalar Zp, we have x+yS and xS.
Two observations:
A subspace S of Znp will always contains 0. Because 0x=0 must be in it.
All nonempty subsets of Zn2 are closed under scalar multiplication.

6. Examples on Subspaces (See the textbook)
Example {0000,0100,0010,0110} is a subspace of Z42.
Example {0000,0001,1000,0110} is not a subspace of Z42.
Example {000,100,200,001,002,101,102,201,202} is a subspace of Z33.
Example {0} and Znp are subspaces of Znp.

7. Linear Combinations
Definition Let x1,x2,…xk be strings in Znp, and let 1, 2,…, k be scalars (i.e.
elements of Zp). The string 1x1+ 2x2…+ kxk is called a linear combination of
the strings x1,x2,…,xk, with coefficients 1, 2,…, k. If all of the coefficients are 0,
then the linear combination is said to be trivial, otherwise it is nontrivial.
Example In Z35, the string x=2(010)+121+3(140)=411 is a (nontrivial) linear
combination of 010, 121 and 140, with coefficients 2,1 and 3.
The elements of the subspace
S1={000,100,200,001,002,101,102,201,202}
of Z33 (Example 5.1.4) can be described in terms linear combinations as
S1={(100)+(001) | ,Z3}.
Because of this, we say that {100,001} generates S1.

8. Generating Sets
Definition Let S be a subspace of Znp. A subset G={g1,…,gk} of S is called a
generating set for S if, for any xS, there exist scalars 1,…,k such that
x = 1g1 +… + kgk.
To indicate that G is a generating set for S, we write S=<G>.
Example {0000,0100,0010,0110}=<0100,0010> (see the textbook).
Theorem Let G be a nonempty set of strings in Znp. The set of all linear
combinations of strings in G is a subspace of Znp, called the subspace generated by G,
or the subspace spanned by G, and denoted by <G>.
Exercise: See that both G1={100,001} and G2={100,001,101} are generating sets for
S={000,100,001,101}.
G2, however, is a wasteful way to describe S, because it is not minimal.

9. Bases
Definition Let S be a subspace of Znp. A generating set B of S is said to be a basis
for S if it is a minimal generating set, in the sense that no proper subset of B
generates S.
Example B={100,001} is a basis for S={000,100,001,101}. Indeed:
B does generate S, because 000 is the trivial combination (i.e. 0(100)+0(001));
100=1(100) (i.e. 1(100)+0(001));
001=1(001) (i.e. 0(100)+1(001);
101= (i.e. 1(100)+1(001)).
No proper subset of B can generate S. Say, {100} does not generate S because
001 is not a scalar multiple of 100. Similarly for {001}.
We are going to see that bases allow us to describe subspaces (=linear codes) without
listing all of the elements.

10. Standard Bases; Dimension
Example The set B={e1,e2,e3,e4}={1000,0100,0010,0001} is a basis for Z42.
Because any string x = u1u2u3u4 in Z42 can be written as
x = u1(1000) + u2(0100) + u3(0010) + u4(0001).
More generally: Let ei denote the string in Znp with 0s in every position except the
ith position, where it has a 1. Then B={e1,…,en} is a basis for Znp. It is called
the standard basis of Znp.
Theorem Let S be a subspace of Znp. Then all bases for S have the same size.
This size is called the dimension of S and is denoted by dim(S).

11. Theory of Information Lecture 13
Homework
Exercises 3, 6, 8, 9, 12, 13.
Remember (and understand) all of the items of this lecture marked as Definition or
Theorem.